% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
%
% Trinity College, 2000.

\magnification=\magstep1
\vsize=227 true mm \hsize=170 true mm
   \voffset=-0.4 true mm \hoffset=-5.4 true mm

\def\folio{\ifnum\pageno>0 \number\pageno \else
   \ifnum\pageno<0 \romannumeral-\pageno \else\fi\fi}

\def\multieqalign#1{\null\,\vcenter{\openup1\jot \mathsurround=0pt
   \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
   &&\quad \strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
    \crcr#1\crcr}}\,}

\font\Largebf=cmbx10  scaled \magstep2
\font\largerm=cmr12
\font\largeit=cmti12
\font\tensc=cmcsc10
\font\sevensc=cmcsc10 scaled 700
\newfam\scfam \def\sc{\fam\scfam\tensc}
\textfont\scfam=\tensc \scriptfont\scfam=\sevensc
\font\ninerm=cmr9
\newfam\srfam \def\sr{\fam\srfam\ninerm}
\textfont\srfam=\ninerm \scriptfont\srfam=\sevenrm
\scriptscriptfont\srfam=\fiverm

\def\overcomma{\mathaccent"7027}
\def\overrevcomma{\mathaccent"7060}
\def\doubleacute{\mathaccent"707D}

\input amssym.def
\newsymbol\backprime 1038

\def\oppos{\mathop{\ominus}\nolimits}

\def\recip{\mathchoice{\mathop{\lower0.05em\hbox{$\textstyle {}^\backprime$}
               \kern-0.20em
   {\rm c} \kern-0.135em \vrule depth0ex height1.63ex width0.09em
               \,}\nolimits}{
                       \mathop{\lower0.05em\hbox{$\textstyle {}^\backprime$}
               \kern-0.20em
   {\rm c} \kern-0.135em \vrule depth0ex height1.63ex width0.09em
               \,}\nolimits}{
                       \mathop{\lower0.5pt\hbox{$\scriptstyle {}^\backprime$}
               \kern-0.23em
   {\rm c} \kern-0.110em \vrule depth0ex height1.25ex width0.06em
               \,}\nolimits}{
                       \mathop{\lower0.5pt\hbox{$\scriptstyle {}^\backprime$}
               \kern-0.23em
   {\rm c} \kern-0.110em \vrule depth0ex height1.25ex width0.06em
               \,}\nolimits}}

\def\neq{\mathrel{\mid\kern-.5em=}}
\def\nlt{\mathrel{\mid\kern-.55em<}}
\def\ngt{\mathrel{\mid\kern-.45em>}}

\def\gteqlt{\mathrel{\mathop{=}\limits^>_<}}
\def\lteqgt{\mathrel{\mathop{=}\limits^<_>}}
\def\leq{\mathrel{\mathop{<}\limits_=}}
\def\geq{\mathrel{\mathop{>}\limits_=}}

\def\subzero#1{\mathord{\mathop{#1}\limits_0}}
\def\putsub#1#2{\mathord{\mathop{#2}\limits_{#1}}}

\pageno=0

\null\vskip72pt

\centerline{\Largebf THEORY OF CONJUGATE FUNCTIONS,}

\vskip12pt

\centerline{\Largebf OR ALGEBRAIC COUPLES;}

\vskip12pt

\centerline{\Largebf WITH A PRELIMINARY AND ELEMENTARY}

\vskip12pt

\centerline{\Largebf ESSAY ON ALGEBRA AS THE}

\vskip12pt

\centerline{\Largebf SCIENCE OF PURE TIME}

\vskip24pt

\centerline{\Largebf By}

\vskip24pt

\centerline{\Largebf William Rowan Hamilton}

\vskip24pt

\centerline{\largerm (Transactions of the Royal Irish Academy, vol.~17,
part~1 (1837), pp. 293--422.)}

\vskip36pt

\vfill

\centerline{\largerm Edited by David R. Wilkins}

\vskip 12pt

\centerline{\largerm 2000}

\vskip36pt\eject

\pageno=-1

\null\vskip36pt

\centerline{\Largebf NOTE ON THE TEXT}

\bigskip

This edition follows the original text in volume 17 of the
{\it Transactions of the Royal Irish Academy}, published in
1837.  In particular, the typographical conventions specified
by Hamilton in article 34 have been followed.

A sentence in parentheses in article 30 has been suppressed which
merely drew attention to a typographical error earlier in the
original text.

A footnote to article 34 describing the preferred visual appearance
of the symbol~$\oppos$ denoting negation has been suppressed.

The glyph $\recip$ has been created to denote the operation of
passing from a number to its reciprocal.  This is intended to
represent an inverted letter~${\rm R}$, as specified by Hamilton
in article 17: `` the initial letter of the word Reciprocatio,
distinguished from the other uses of this same letter by being
written in an inverted position''.

The capital letter~${\rm B}$ has been used in article~32 to
denote a positive ratio.  (The original text uses the small
capital letter ${\sc b}$, which is used systematically elsewhere
in the essay to denote moments in time.)

A number of small typographical errors noted by Hamilton in a
published list of errata have been corrected without comment,
as have a small number of errors of a similar nature.

\bigbreak\bigskip

\line{\hfil David R. Wilkins}

\vskip3pt

\line{\hfil Dublin, March 2000}

\vfill\eject

\pageno=1

\null\vskip36pt

{\largeit\noindent
Theory of Conjugate Functions, or Algebraic Couples; with a
Preliminary and Elementary Essay on Algebra as the Science
of Pure Time.}

\bigskip

\centerline{\largerm By WILLIAM ROWAN HAMILTON,}

\bigskip

{\largeit\noindent
M.R.I.A., F.R.A.S., Hon.\ M. R. S. Ed.\ and Dub.,
Fellow of the American Academy of Arts and Sciences, and of the
Royal Northern Antiquarian Society at Copenhagen,
Andrews' Professor of Astronomy in the University of Dublin, and
Royal Astronomer of Ireland.}

\bigbreak

\centerline{Read November 4th, 1833, and June 1st, 1835.}

\bigbreak

\centerline{[{\it Transactions of the Royal Irish Academy},
vol.~17 (1837), pp. 293--422.]}

\nobreak\bigskip

\centerline{\vbox{\hrule width 72pt}}

\nobreak\bigskip

\centerline{\largeit General Introductory Remarks.}

\nobreak\bigskip

The Study of Algebra may be pursued in three very different
schools, the Practical, the Philological, or the Theoretical,
according as Algebra itself is accounted an Instrument, or a
Language, or a Contemplation; according as ease of operation, or
symmetry of expression, or clearness of thought, (the
{\it agere}, the {\it fari}, or the {\it sapere},) is eminently
prized and sought for.  The Practical person seeks a Rule which
he may apply, the Philological person seeks a Formula which he
may write, the Theoretical person seeks a Theorem on which he may
meditate.  The felt imperfections of Algebra are of three
answering kinds.  The Practical Algebraist complains of
imperfection when he finds his Instrument limited in power; when
a rule, which he could happily apply to many cases, can be hardly
or not at all applied by him to some new case; when it fails to
enable him to do or to discover something else, in some other
Art, or in some other Science, to which Algebra with him was but
subordinate, and for the sake of which and not for its own sake,
he studied Algebra.  The Philological Algebraist complains of
imperfection, when his Language presents him with an Anomaly;
when he finds an Exception disturb the simplicity of his
Notation, or the symmetrical structure of his Syntax; when a
Formula must be written with precaution, and a Symbolism is not
universal.  The Theoretical Algebraist complains of imperfection,
when the clearness of his Contemplation is obscured; when the
Reasonings of his Science seem anywhere to oppose each other, or
become in any part too complex or too little valid for his belief
to rest firmly upon them; or when, though trial may have taught
him that a rule is useful, or that a formula gives true results,
he cannot prove that rule, nor understand that formula: when he
cannot rise to intuition from induction, or cannot look beyond
the signs to the things signified.

It is not here asserted that every or any Algebraist belongs
{\it exclusively\/} to any {\it one\/} of these three schools, so
as to be {\it only\/} Practical, or {\it only\/} Philological, or
{\it only\/} Theoretical.  Language and Thought react, and
Theory and Practice help each other.  No man can be so merely
practical as to use frequently the rules of Algebra, and never to
admire the beauty of the language which expresses those rules,
nor care to know the reasoning which deduces them.  No man can be
so merely philological an Algebraist but that things or thoughts
will at some times intrude upon signs; and occupied as he may
habitually be with the logical building up of his expressions, he
will feel sometimes a desire to know what they mean, or to apply
them.  And no man can be so merely theoretical or so exclusively
devoted to thoughts, and to the contemplation of theorems in
Algebra, as not to feel an interest in its notation and language,
its symmetrical system of signs, and the logical forms of their
combinations; or not to prize those practical aids, and
especially those methods of research, which the discoveries and
contemplations of Algebra have given to other sciences.  But,
distinguishing without dividing, it is perhaps correct to say
that every Algebraical Student and every Algebraical Composition
may be referred upon the whole to one or other of these three
schools, according as one or other of these three views
habitually actuates the man, and eminently marks the work.

These remarks have been premised, that the reader may more easily
and distinctly perceive what the design of the following
communication is, and what the Author hopes or at least desires
to accomplish.  That design is {\it Theoretical}, in the sense
already explained, as distinguished from what is Practical on the
one hand, and what is Philological upon the other.  The thing
aimed at, is to improve the {\it Science}, not the Art nor the
Language of Algebra.  The imperfections sought to be removed, are
confusions of thought, and obscurities or errors of reasoning;
not difficulties of application of an instrument, nor failures of
symmetry in expression.  And that confusions of thought, and
errors of reasoning, still darken the beginnings of Algebra, is
the earnest and just complaint of sober and thoughtful men, who
in a spirit of love and honour have studied Algebraic Science,
admiring, extending, and applying what has been already brought
to light, and feeling all the beauty and consistence of many a
remote deduction, from principles which yet remain obscure, and
doubtful.

For it has not fared with the principles of Algebra as with the
principles of Geometry.  No candid and intelligent person can
doubt the truth of the chief properties of {\it Parallel Lines},
as set forth by {\sc Euclid} in his Elements, two thousand years
ago; though he may well desire to see them treated in a clearer
and better method.  The doctrine involves no obscurity nor
confusion of thought, and leaves in the mind no reasonable ground
for doubt, although ingenuity may usefully be exercised in
improving the plan of the argument.  But it requires no peculiar
scepticism to doubt, or even to disbelieve, the doctrine of
Negatives and Imaginaries, when set forth (as it has commonly
been) with principles like these: that a {\it greater magnitude
may be subtracted from a less}, and that the remainder is
{\it less than nothing\/}; that {\it two negative numbers}, or
numbers denoting magnitudes each less than nothing, may be
{\it multiplied\/} the one by the other, and that the product
will be a {\it positive\/} number, or a number denoting a
magnitude greater than nothing; and that although the
{\it square\/} of a number, or the product obtained by
multiplying that number by itself, is therefore {\it always
positive}, whether the number be positive or negative, yet that
numbers, called {\it imaginary}, can be found or conceived or
determined, and operated on by all the rules of positive and
negative numbers, as if they were subject to those rules,
{\it although they have negative squares}, and must therefore be
supposed to be themselves neither positive nor negative, nor yet
null numbers, so that the magnitudes which they are supposed to
denote can neither be greater than nothing, nor less than
nothing, nor even equal to nothing.  It must be hard to found a
{\sc Science\/} on such grounds as these, though the forms of
logic may build up from them a symmetrical system of expressions,
and a practical art may be learned of rightly applying useful
rules which seem to depend upon them.

So useful are those rules, so symmetrical those expressions, and
yet so unsatisfactory those principles from which they are
supposed to be derived, that a growing tendency may be perceived
to the rejection of that view which regarded Algebra as a
{\sc Science}, {\it in some sense analogous to Geometry}, and to
the adoption of one or other of those two different views, which
regard Algebra as an {\it Art}, or as a {\it Language\/}: as a
System of Rules, or else as a System of Expressions, but not as a
System of {\it Truths}, or Results having any other validity than
what they may derive from their practical usefulness, or their
logical or philological coherence.  Opinions thus are tending to
substitute for the Theoretical question,---``Is a Theorem of
Algebra {\it true\/}?'' the Practical question,---``Can it be
{\it applied as an Instrument}, to do or to discover something
else, in some research which is not Algebraical?'' or else the
Philological question,---``Does its {\it expression harmonise},
according to the Laws of Language, with other Algebraical
expressions?''

Yet a natural regret might be felt, if such were the destiny of
Algebra; if a study, which is continually engaging mathematicians
more and more, and has almost superseded the Study of Geometrical
Science, were found at last to be not, in any strict or proper
sense, the Study of a Science at all: and if, in thus exchanging
the ancient for the modern Mathesis, there were a gain only of
Skill or Elegance, at the expense of Contemplation and Intuition.
Indulgence, therefore, may be hoped for, by any one who would
inquire, whether existing Algebra, in the state to which it has
been already unfolded by the masters of its rules and of its
language, offers indeed no rudiment which may encourage a hope
of developing a {\sc Science\/} of Algebra: a Science properly so
called: strict, pure and independent; deduced by valid reasonings
from its own intuitive principles; and thus not less an object of
priori contemplation than Geometry, nor less distinct, in its
own essence, from the Rules which it may teach or use, and from
the Signs by which it may express its meaning.

The Author of this paper has been led to the belief, that the
Intuition of {\sc Time\/} is such a rudiment.

This belief involves the three following as components: First,
that the notion of Time is connected with existing Algebra;
Second, that this notion or intuition of Time may be unfolded
into an independent Pure Science; and Third, that the Science of
Pure Time, thus unfolded, is co-extensive and identical with
Algebra, so far as Algebra itself is a Science.  The first
component judgment is the result of an induction; the second of a
deduction; the third is the joint result of the deductive and
inductive processes.

\bigbreak

I.
The argument for the conclusion that {\it the notion of Time is
connected with existing Algebra}, is an induction of the
following kind.  The History of Algebraic Science shows that the
most remarkable discoveries in it have been made, either
expressly through the medium of that notion of {\it Time}, or
through the closely connected (and in some sort coincident)
notion of {\it Continuous Progression}.  It is the genius of
Algebra to consider what it reasons on as {\it flowing}, as it
was the genius of Geometry to consider what it reasoned on as
{\it fixed}.  {\sc Euclid}\footnote*{%
$E \overcomma{\upsilon} \theta \epsilon \tilde{\iota} \alpha$
$\kappa \acute{\upsilon} \kappa \lambda o \upsilon$
$\overcomma{\epsilon} \phi \acute{\alpha} \pi \tau \epsilon
   \sigma \theta \alpha \iota$
$\lambda \acute{\epsilon} \gamma \epsilon \tau \alpha \iota$,
$\doubleacute{\eta} \tau \iota \varsigma$
$\overrevcomma{\alpha} \pi \tau o \mu \acute{\epsilon} \nu \eta$
$\tau o \tilde{\upsilon}$
$\kappa \acute{\upsilon} \kappa \lambda o \nu$
$\kappa \alpha \grave{\iota}$
$\overcomma{\epsilon} \kappa \beta \alpha \lambda \lambda o \mu
   \acute{\epsilon} \nu \eta$
$o \overcomma{\upsilon}$
$\tau \acute{\epsilon} \mu \nu \epsilon \iota$
$\tau \grave{o} \nu$
$\kappa \acute{\upsilon} \kappa \lambda o \nu$.---{\sc Euclid},
Book~{\sc iii}. Def.~2.  Oxford Edition, 1703.}
defined a tangent to a circle,  {\sc Apollonius}\footnote*{%
$E\grave{\alpha}\nu$
$\overcomma{\epsilon} \nu$
$\kappa \acute{\omega} \nu o \upsilon$
$\tau o \mu \tilde{\eta}$
$\overcomma{\alpha} \pi \grave{o}$
$\tau \tilde{\eta} \varsigma$
$\kappa o \rho \upsilon \phi \tilde{\eta} \varsigma$
$\tau \tilde{\eta} \varsigma$
$\tau o \mu \tilde{\eta} \varsigma$
$\overcomma{\alpha} \chi \theta \tilde{\eta}$
$\epsilon \overcomma{\upsilon} \theta \epsilon \tilde{\iota} \alpha$
$\pi \alpha \rho \grave{\alpha}$
$\tau \epsilon \tau \alpha \gamma \mu \acute{\epsilon} \nu \omega
   \varsigma$
$\kappa \alpha \tau \eta \gamma \mu \acute{\epsilon} \nu \eta
   \nu$
$\overcomma{\epsilon} \kappa \tau \grave{o} \varsigma$
$\pi \epsilon \sigma \epsilon \tilde{\iota} \tau \alpha \iota$
$\tau \tilde{\eta} \varsigma$
$\tau o \mu \tilde{\eta} \sigma$.---$\overcomma{\epsilon} \kappa
   \tau \grave{o} \varsigma$
$\doubleacute{\alpha} \rho \alpha$
$\pi \epsilon \sigma \epsilon \tilde{\iota} \tau \alpha \iota$,
$\delta \iota \acute{o} \pi \epsilon \rho$
$\overcomma{\epsilon} \phi \acute{\alpha} \pi \tau \epsilon \tau
   \alpha \iota$
$\tau \tilde{\eta} \varsigma$ $\tau o \mu \tilde{\eta}
\varsigma$. ---{\sc Apollonius}, Book~{\sc i}. Prop.~17.  Oxford
Edition, 1710.}
conceived a tangent to an ellipse, as an indefinite straight line
which had only one point in common with the curve; they looked
upon the line and curve not as nascent or growing, but as already
constructed and existing in space; they studied them as
{\it formed\/} and {\it fixed}, they compared the one with the
other, and the proved exclusion of any second common point was to
them the essential property, the constitutive character of the
tangent.  The Newtonian Method of Tangents rests on another
principle; it regards the curve and line not as {\it already\/}
formed and fixed, but rather as {\it nascent}, or in process of
generation: and employs, as its primary conception, the thought
of a {\it flowing point}.  And, generally, the revolution which
{\sc Newton}\footnote\dag{Considerando igitur quod quantitates
{\ae}qualibus temporibus crescentes et crescendo genitae, pro
velocitate majori vel minori qua crescunt ac generantur evadunt
majores vel minores; methodum qu{\ae}rebam determinandi
quantitates ex velocitatibus motuum vel incrementorum quibus
generantur; et has motuum vel incrementorum velocitates nominando
{\it Fluxiones}, et quantitates genitas nominando {\it Fluentes},
incidi paulatim annis 1665 et 1666 in Methodum Fluxionum qua hic
usus sum in Quadratura Curvarum---{\it Tractatus de
Quad.\ Curv.}, Introd., published at the end of Sir I. Newton's
Opticks, London 1704.}
made in the higher parts of both pure and applied Algebra, was
founded mainly on the notion of {\it fluxion}, which involves the
notion of {\it Time}.

Before the age of {\sc Newton}, another great revolution, in
Algebra as well as in Arithmetic, had been made by the invention
of {\it Logarithms\/}; and the ``Canon Mirificus'' attests that
{\sc Napier}\footnote\ddag{Logarithmus erg\`{o} cujusque sinus,
est numerus quam proxim\`{e} definiens lineam, qu{\ae}
{\ae}qualiter crevit intere\`{a} dum sinus totius linea
proportionaliter in sinum illum decrevit, existente utroque motu
synchrono, atque initio {\ae}quiveloce.  Baron Napier's
{\it Mirifici Logarithmorum Canonis Descriptio}, Def.~6,
Edinburgh 1614.---Also in the explanation of Def.~1, the words
{\it fluxu\/} and {\it fluat\/} occur.}
deduced that invention, not (as it is commonly said) from the
arithmetical properties of powers of numbers, but from the
contemplation of a {\it Continuous Progression\/}; in describing
which, he speaks expressly of {\it Fluxions}, {\it Velocities\/}
and {\it Times}.

In a more modern age, {\sc Lagrange}, in the Philological spirit,
sought to reduce the Theory of Fluxions to a system of operations
upon symbols, analogous to the earliest symbolic operations of
Algebra, and professed to reject the notion of time as foreign to
such a system; yet admitted\footnote\S{Calcul des Fonctions,
Le\c{c}on Premiere, page~2.  Paris 1806.}
that fluxions might be considered only as the velocities with
which magnitudes vary, and that in so considering them,
abstraction might be made of every mechanical idea.  And in one
of his own most important researches in pure Algebra, (the
investigation of limits between which the sum of any number of
terms in {\sc Taylor}'s Series is comprised,)
{\sc Lagrange}\footnote{$\|$}{Donc puisque $V$ devient nul
lorsque $i$ devient nul, il est clair qu'en faisant
cro\^{\i}tre~$i$ par degr\'{e}s insensibles depuis z\'{e}ro, la
valeur de $V$ cro\^{\i}tra aussi insensiblement depuis z\'{e}ro,
soit en plus ou en moins, jusqu' \`{a} un certain point,
apr\`{e}s quoi elle pourra diminuer.---Calcul des Fonctions,
Le\c{c}on Neuvi\`{e}me, page 90.  Paris 1806.  An instance still
more strong may be found in the First Note to Lagrange's
{\it Equations Numeriques}.  Paris, 1808.}
employs the conception of {\it continuous progression\/} to show
that a certain variable quantity may be made as small as can be
desired.  And not to dwell on the beautiful discoveries made by
the same great mathematician, in the theory of singular
primitives of equations, and in the algebraical dynamics of the
heavens, through an extension of the conception of
{\it variability}, (that is, in fact, of {\it flowingness},) to
quantities which had before been viewed as {\it fixed\/} or
constant, it may suffice for the present to observe that
{\sc Lagrange} considered Algebra to be the {\it Science of
Functions},\footnote*{On doit regarder l'alg\`{e}bre comme la
science des fonctions.---Calc.\ des Fonct., Le\c{c}on Premiere.}
and that it is not easy to conceive a clearer or juster idea of a
{\it Function} in this Science, than by regarding its essence as
consisting in a {\it Law connecting Change with Change}.  But
where {\it Change\/} and {\it Progression\/} are, there is
{\sc Time}.  The notion of Time is, therefore, inductively found
to be connected with existing Algebra.\footnote\dag{The word
``Algebra'' is used throughout this whole paper, in the sense
which is commonly but improperly given by modern mathematical
writers to the name ``Analysis,'' and not with that narrow
signification to which the unphilosophical use of the latter term
(Analysis) has caused the former term (Algebra) to be too
commonly confined.  The author confesses that he has often
deserved the censure which he has here so freely expressed.}

\bigbreak

II.
The argument for the conclusion that {\it the notion of time may
be unfolded into an independent Pure Science}, or that a
{\it Science of Pure Time is possible}, rests chiefly on the
existence of certain a priori intuitions, connected with that
notion of time, and fitted to become the sources of a pure
Science; and on the actual deduction of such a Science from those
principles, which the author conceives that he has begun.
Whether he has at all succeeded in {\it actually effecting\/}
this deduction, will be judged after the Essay has been read; but
that such a deduction is {\it possible}, may be concluded in an
easier way, by an appeal to those intuitions to which allusion
has been made.  That a moment of time respecting which we
inquire, as compared with a moment which we know, must either
coincide with or precede or follow it, is an intuitive truth, as
certain, as clear, and as unempirical as this, that no two
straight lines can comprehend an area.  The notion or intuition
of {\sc Order in Time} is not less but more deep-seated in the
human mind, than the notion of intuition of {\sc Order in
Space}; and a mathematical Science may be founded on the
former, as pure and as demonstrative as the science founded on
the latter.  There is something mysterious and transcendent
involved in the idea of Time; but there is also something
definite and clear: and while Metaphysicians meditate on the one,
Mathematicians may reason from the other.

\bigbreak

III.
That the {\it Mathematical Science of Time}, when sufficiently
unfolded, and distinguished on the one hand from all actual
Outward Chronology (of collections of recorded events and
phenomenal marks and measures), and on the other hand from all
Dynamical Science (or reasonings and results from the notion of
cause and effect), will ultimately be found to be co-extensive
and identical with Algebra, so far as Algebra itself is a
Science: is a conclusion to which the author has been led by all
his attempts, whether to {\it analyse\/} what is
{\it Scientific in Algebra}, or to {\it construct a Science of
Pure Time}.  It is a joint result of the inductive and deductive
processes, and the grounds on which it rests could not be stated
in a few general remarks.  The author hopes to explain them more
fully in a future paper; meanwhile he refers to the present one,
as removing (in his opinion) the difficulties of the usual theory
of Negative and Imaginary Quantities, or rather substituting a
new Theory of {\it Contrapositives\/} and {\it Couples}, which he
considers free from those old difficulties, and which is deduced
from the Intuition or Original Mental Form of Time: the
opposition of the (so-called) Negatives and Positives being
referred by him, {\it not\/} to the opposition of the operations
of increasing and diminishing a {\it magnitude}, but to the
simpler and more extensive contrast between the relations of
{\it Before\/} and {\it After\/},\footnote*{It is, indeed, very
common, in Elementary works upon Algebra, to allude to {\it past
and future time}, as one among many {\it illustrations\/} of the
doctrine of negative quantities; but this avails little for
Science, so long as {\it magnitude\/} instead of
{\sc progression} is attempted to be made the {\it basis\/} of
the doctrine.}
or between the directions of {\it Forward\/} and
{\it Backward\/}; and {\it Pairs of Moments\/} being used to
suggest a {\it Theory of Conjugate Functions},\footnote\dag{The
author was conducted to this Theory many years ago, in reflecting
on the important symbolic results of Mr.~{\sc Graves} respecting
Imaginary Logarithms, and in attempting to explain to himself the
theoretical meaning of those remarkable symbolisms.  The
Preliminary and Elementary Essay of Algebra as the Science of
Pure Time, is a much more recent development of an Idea against
which the author struggled long, and which he still longer
forbore to make public, on account of its departing so far from
views now commonly received.  The novelty, however, is in the
view and method, not in the results and details: in which the
reader is warned to expect little addition, if any, to what is
already known.}
which gives reality and meaning to conceptions that were before
Imaginary,\footnote\ddag{The author acknowledges with pleasure
that he agrees with M.~{\sc Cauchy}, in considering every
(so-called) Imaginary Equation as a symbolic representation of
two separate Real Equations: but he differs from that excellent
mathematician in his method generally, and especially in not
introducing the sign $\surd - 1$ until he has provided for it, by
his Theory of Couples, a possible and real meaning, as a symbol
of the couple $(0,1)$.}
Impossible, or Contradictory, because Mathematicians had derived
them from that bounded notion of {\it Magnitude}, instead of the
original and comprehensive thought of {\sc Order in Progression}.

\vfill\eject

\centerline{\largerm CONTENTS OF THE PRELIMINARY AND ELEMENTARY}

\vskip 6pt

\centerline{\largerm ESSAY ON ALGEBRA AS THE SCIENCE OF PURE TIME.}

\bigskip

\centerline{\vbox{\hrule width 72pt}}

\begingroup
\everypar{\parindent=0pt \hangindent=20pt \hangafter=1}

\nobreak\bigskip

\line{\hfil Articles}

\nobreak\bigskip

\noindent
Comparison of any two moments with respect to identity or
diversity, subsequence or precedence.
\dotfill \hbox{1}

\smallskip

Comparison of two pairs of moments, with respect to their
analogy or non-analogy.
\dotfill \hbox{2, 3}

\smallskip

Combinations of two different analogies, or non-analogies, of
pairs of moments, with each other.
\dotfill \hbox{4}

\smallskip

On continued analogies, or equidistant series of moments.
\dotfill \hbox{5, 6, 7, 8}

\smallskip

On steps in the progression of time: their application direct or
inverse to moments, so as to generate other moments; and their
combination with other steps, in the way of composition or
decomposition.
\dotfill \hbox{9, 10, 11, 12}

\smallskip

On the multiples of a given base, or unit-step; and on the
algebraic addition, subtraction, multiplication, and division, of
their determining or multipling whole numbers, whether positive
or contra-positive, or null.
\dotfill \hbox{13, 14, 15}

\smallskip

On the submultiples and fractions of any given step in the
progression of time; on the algebraic addition, subtraction,
multiplication, and division, of reciprocal and fractional
numbers, positive and contra-positive; and on the impossible or
indeterminate act of submultipling or dividing by zero.
\dotfill \hbox{16, 17, 18, 19, 20}

\smallskip

On the comparison of any one effective step with any other, in
the way of ratio, and the generation of any one such step from
any other, in the way of multiplication; and on the addition,
subtraction, multiplication, and division of algebraic numbers in
general, considered thus as ratios or as multipliers of steps.
\dotfill \hbox{21, 22, 23}

\smallskip

On the insertion of a mean proportional between two steps; and on
impossible, ambiguous, and incommensurable square roots of
ratios.
\dotfill \hbox{24, 25}

\smallskip

More formal proof of the general existence of a determined
positive square-root, commensurable or incommensurable, for every
determined positive ratio; continuity of progression of the
square, and principles connected with this continuity.
\dotfill \hbox{26, 27}

\smallskip

On continued analogies or series of proportional steps; and on
powers and roots and logarithms of ratios.
\dotfill \hbox{28, 29, 30, 31, 32, 33}

\smallskip

Remarks on the notation of this Essay and on some modifications
by which it may be made more like the notation commonly employed.
\dotfill \hbox{34, 35, 36}

\endgroup

\vfill\eject

\centerline{\largerm CONTENTS OF THE THEORY OF CONJUGATE FUNCTIONS,}

\vskip 6pt

\centerline{\largerm OR ALGEBRAIC COUPLES.}

\bigskip

\centerline{\vbox{\hrule width 72pt}}

\begingroup
\everypar{\parindent=0pt \hangindent=20pt \hangafter=1}

\nobreak\bigskip

\line{\hfil Articles}

\nobreak\bigskip

\noindent
On couples of moments, and of steps, in time.
\dotfill \hbox{1}

\smallskip

On the composition and decomposition of step-couples.
\dotfill \hbox{2}

\smallskip

On the multiplication of a step-couple by a number.
\dotfill \hbox{3}

\smallskip

On the multiplication of a step-couple by a number-couple; and on
the ratio of one step-couple to another.
\dotfill \hbox{4, 5}

\smallskip

On the addition, subtraction, multiplication, and division, of
number-couples, as combined with each other.
\dotfill \hbox{6}

\smallskip

On the powering of a number-couple by a single whole number.
\dotfill \hbox{7, 8, 9}

\smallskip

On a particular class of exponential and logarithmic
function-couples, connected with a particular series of integer
powers of number-couples.
\dotfill \hbox{10, 11, 12}

\smallskip

On the powering of any number-couple by any single number or
number-couple.
\dotfill \hbox{13, 14, 15}

\smallskip

On exponential and logarithmic function-couples in general.
\dotfill \hbox{16}

\endgroup

\vfill\eject

\centerline{\largerm PRELIMINARY AND ELEMENTARY ESSAY}

\vskip 6pt

\centerline{\largerm ON ALGEBRA AS THE SCIENCE OF PURE TIME.}

\nobreak\bigskip

\centerline{\vbox{\hrule width 72pt}}

\nobreak\bigskip

\centerline{\it Comparison of any two moments with respect to
identity or diversity,}

\nobreak\vskip 3pt

\centerline{\it subsequence or precedence.}

\nobreak\bigskip

1.
If we have formed the {\it thought\/} of any one moment of time,
we may afterwards either {\it repeat\/} that thought, or else
think of a {\it different\/} moment.  And if any two spoken or
written {\it names}, such as the letters ${\sc a}$ and ${\sc b}$,
be {\it dates}, or answers to the question {\it When}, denoting
each a known moment of time, they must either be names of one and
the {\it same\/} known moment, or else of two {\it different\/}
moments.  In each case, we may speak of the {\it pair of dates\/}
as denoting a {\it pair of moments\/}; but in the first case, the
two moments are coincident, while in the second case they are
distinct from each other.  To express concisely the former case
of relation, that is, the case of {\it identity\/} between the
moment named ${\sc b}$ and the moment named ${\sc a}$, or of
{\it equivalence\/} between the date~${\sc b}$ and the
date~${\sc a}$, it is usual to write
$${\sc b} = {\sc a};
   \eqno (1.)$$
a written sentence or assertion, which is commonly called an
{\it equation\/}: and to express concisely the latter case of
relation, that is, the case of {\it diversity\/} between the two
moments, or of {\it non-equivalence\/} between the two dates, we
may write
$${\sc b} \neq {\sc a};
   \eqno (2.)$$
annexing, here and afterwards, to these concise written
expressions, the side-marks (1.) (2.), \&c., merely to facilitate
the subsequent reference in this essay to any such assertion or
result, whenever such reference may become necessary or
convenient.  The latter case of relation, namely, the case (2.)
of diversity between two moments, or of non-equivalence between
two dates, subdivides itself into the two cases of
{\it subsequence\/} and of {\it precedence}, according as the
moment~${\sc b}$ is later or earlier than~${\sc a}$.  To express
concisely the former sort of diversity, in which the moment
${\sc b}$ is {\it later\/} than ${\sc a}$, we may write
$${\sc b} > {\sc a};
   \eqno (3.)$$
and the latter sort of diversity, in which the moment~${\sc b}$
is {\it earlier\/} than ${\sc a}$, may be expressed concisely in
this other way,
$${\sc b} < {\sc a}.
   \eqno (4.)$$
It is evident that
$$\eqalignno{
\hbox{if}\quad {\sc b} = {\sc a},\quad \hbox{then}\quad
   {\sc a} = {\sc b}; &&(5.)\cr
\hbox{if}\quad {\sc b} \neq {\sc a},\quad \hbox{then}\quad
   {\sc a} \neq {\sc b}; &&(6.)\cr
\hbox{if}\quad {\sc b} > {\sc a},\quad \hbox{then}\quad
   {\sc a} < {\sc b}; &&(7.)\cr
\hbox{if}\quad {\sc b} < {\sc a},\quad \hbox{then}\quad
   {\sc a} > {\sc b}. &&(8.)\cr}$$

\bigbreak

\centerline{\it
Comparison of two pairs of moments, with respect to their
analogy or non-analogy.}

\nobreak\bigskip

2.
Considering now any two other dates ${\sc c}$ and ${\sc d}$, we
perceive that they may and must represent either the {\it same\/}
pair of moments as that denoted by the former pair of dates
${\sc a}$ and ${\sc b}$, or else a {\it different\/} pair,
according as the two conditions,
$${\sc c} = {\sc a},\quad \hbox{and}\quad {\sc d} = {\sc b},
   \eqno (9.)$$
are, or are not, both satisfied.  If the new pair of moments be
the {\it same\/} with the old, then the connecting relation of
identity or diversity between the moments of the one pair is
necessarily the same with the relation which connects in like
manner the moments of the other pair, because the pairs
themselves are the same.  But even if the {\it pairs\/} be
{\it different}, the {\it relations\/} may still be the same;
that is, the moments ${\sc c}$ and ${\sc d}$, even if not both
respectively coincident with the moments ${\sc a}$ and ${\sc b}$,
may still be related to each other exactly as those moments,
(${\sc d}$ to ${\sc c}$ as ${\sc b}$ to ${\sc a}$;) and thus the
two pairs ${\sc a}, {\sc b}$ and ${\sc c}, {\sc d}$ may be
{\it analogous}, even if they be {\it not coincident\/} with each
other.  An {\it analogy\/} of this sort (whether between
coincident or different pairs) may be expressed in writing as
follows,
$${\sc d} - {\sc c} = {\sc b} - {\sc a},
   \quad\hbox{or}\quad
  {\sc b} - {\sc a} = {\sc d} - {\sc c};
   \eqno (10.)$$
the interposed mark~$=$, which before denoted identity of
moments, denoting now identity of relations: and the written
assertion of this identity being called (as before) an
{\it equation}.  The conditions of this exact identity between
the relation of the moment ${\sc d}$ to ${\sc c}$, and that of
${\sc b}$ to ${\sc a}$, may be stated more fully as follows: that
if the moment ${\sc b}$ be identical with ${\sc a}$, then
${\sc d}$ must be identical with ${\sc c}$; if ${\sc b}$ be later
than ${\sc a}$, then ${\sc d}$ must be later than ${\sc c}$, and
exactly so much later; and if ${\sc b}$ be earlier than
${\sc a}$, then ${\sc d}$ must be earlier than ${\sc c}$ and
exactly so much earlier.  It is evident, that whatever the
moments ${\sc a}$~${\sc b}$ and ${\sc c}$ may be, there is always
one, and only one, connected moment~${\sc d}$, which is thus
related to ${\sc c}$, {\it exactly\/} as ${\sc b}$ is to
${\sc a}$; and it is not difficult to perceive that the same
moment ${\sc d}$ is also related to ${\sc b}$, exactly as
${\sc c}$ is to ${\sc a}$: since, in the case of coincident
pairs, ${\sc d}$ is identical with ${\sc b}$, and ${\sc c}$ with
${\sc a}$; while, in the case of pairs analogous but not
coincident, the moment ${\sc d}$ is earlier or later than
${\sc b}$, according as ${\sc c}$ is later or earlier than
${\sc a}$, and exactly so much later or so much earlier.  If then
the pairs ${\sc a}, {\sc b}$, and ${\sc c}, {\sc d}$, be
analogous, the pairs ${\sc a}$~${\sc c}$ and ${\sc b}$~${\sc d}$,
which may be said to be {\it alternate\/} to the former, are also
analogous pairs; that is,
$$\hbox{if}\quad
   {\sc d} - {\sc c} = {\sc b} - {\sc a},
   \quad\hbox{then}\quad
   {\sc d} - {\sc b} = {\sc c} - {\sc a};
   \eqno (11.)$$
a change of statement of the relation between these four moments
${\sc a}$~${\sc b}$~${\sc c}$~${\sc d}$, which may be called
{\it alternation\/} of an analogy.  It is still more easy to
perceive, that if any two pairs ${\sc a}$~${\sc b}$ and
${\sc c}$~${\sc d}$ be analogous, then the {\it inverse\/} pairs
${\sc b}$~${\sc a}$ and ${\sc d}$~${\sc c}$ are analogous also,
and therefore that
$$\hbox{if}\quad
   {\sc d} - {\sc c} = {\sc b} - {\sc a},
   \quad\hbox{then}\quad
   {\sc c} - {\sc d} = {\sc a} - {\sc b},
   \eqno (12.)$$
a change in the manner of expressing the relation between the
four moments ${\sc a}$~${\sc b}$~${\sc c}$~${\sc d}$, which may
be called {\it inversion\/} of an analogy.  Combining inversion
with alternation, we find that
$$\hbox{if}\quad
   {\sc d} - {\sc c} = {\sc b} - {\sc a},
   \quad\hbox{then}\quad
   {\sc b} - {\sc d} = {\sc a} - {\sc c};
   \eqno (13.)$$
and thus that all the eight following written sentences expresses
only one and the same relation between the four moments
${\sc a}$~${\sc b}$~${\sc c}$~${\sc d}$:
$$\left. \eqalign{
{\sc d} - {\sc c} &= {\sc b} - {\sc a},\quad
{\sc b} - {\sc a}  = {\sc d} - {\sc c},\cr
{\sc d} - {\sc b} &= {\sc c} - {\sc a},\quad
{\sc c} - {\sc a}  = {\sc d} - {\sc b},\cr
{\sc c} - {\sc d} &= {\sc a} - {\sc b},\quad
{\sc a} - {\sc b}  = {\sc c} - {\sc d},\cr
{\sc b} - {\sc d} &= {\sc a} - {\sc c},\quad
{\sc a} - {\sc c}  = {\sc b} - {\sc d};\cr}
   \right\}
   \eqno (14.)$$
any one of these eight written sentences or {\it equations\/}
being equivalent to any other.

\bigbreak

3.
When the foregoing relation between four moments
${\sc a}$~${\sc b}$~${\sc c}$~${\sc d}$ does not exist, that is,
when the pairs ${\sc a}$~${\sc b}$ and ${\sc c}$~${\sc d}$ are
not analogous pairs, we may mark this {\it non-analogy\/} by
writing
$${\sc d} - {\sc c} \neq {\sc b} - {\sc a};
   \eqno (15.)$$
and the two possible cases into which this general conception of
non-analogy or diversity of relation subdivides itself, namely,
the case when the analogy fails on account of the
moment~${\sc d}$ being {\it too late}, and the case when it fails
because that moment ${\sc d}$ is {\it too early}, may be denoted,
respectively, by writing in the first case,
$${\sc d} - {\sc c} > {\sc b} - {\sc a},
   \eqno (16.)$$
and in the second case,
$${\sc d} - {\sc c} < {\sc b} - {\sc a};
   \eqno (17.)$$
while the two cases themselves may be called, respectively, a
{\it non-analogy of subsequence}, and a {\it non-analogy of
precedence}.  We may also say that the relation of ${\sc d}$ to
${\sc c}$, as compared with that of ${\sc b}$ to ${\sc a}$, is in
the first case a relation of {\it comparative lateness}, and in
the second case a relation of {\it comparative earliness}.

Alternations and inversions may be applied to these expressions
of non-analogy, and the case of ${\sc d}$ {\it too late\/} may be
expressed in any one of the eight following ways, which are all
equivalent to each other,
$$\left. \eqalign{
{\sc d} - {\sc c} &> {\sc b} - {\sc a},\quad
{\sc b} - {\sc a}  < {\sc d} - {\sc c},\cr
{\sc d} - {\sc b} &> {\sc c} - {\sc a},\quad
{\sc c} - {\sc a}  < {\sc d} - {\sc b},\cr
{\sc c} - {\sc d} &< {\sc a} - {\sc b},\quad
{\sc a} - {\sc b}  > {\sc c} - {\sc d},\cr
{\sc b} - {\sc d} &< {\sc a} - {\sc c},\quad
{\sc a} - {\sc c}  > {\sc b} - {\sc d};\cr}
   \right\}
   \eqno (18.)$$
while the other case, when the analogy fails because the moment
${\sc d}$ is {\it too early}, may be expressed in any of the
eight ways following,
$$\left. \eqalign{
{\sc d} - {\sc c} &< {\sc b} - {\sc a},\quad
{\sc b} - {\sc a}  > {\sc d} - {\sc c},\cr
{\sc d} - {\sc b} &< {\sc c} - {\sc a},\quad
{\sc c} - {\sc a}  > {\sc d} - {\sc b},\cr
{\sc c} - {\sc d} &> {\sc a} - {\sc b},\quad
{\sc a} - {\sc b}  < {\sc c} - {\sc d},\cr
{\sc b} - {\sc d} &> {\sc a} - {\sc c},\quad
{\sc a} - {\sc c}  < {\sc b} - {\sc d}.\cr}
   \right\}
   \eqno (19.)$$
In general, if we have any analogy or non-analogy between two
pairs of moments, ${\sc a}$~${\sc b}$ and ${\sc c}$~${\sc d}$, of
which we may call the first and fourth mentioned moments,
${\sc a}$ and ${\sc d}$, the {\it extremes}, and the second and
third mentioned moments, namely, ${\sc b}$ and ${\sc c}$, the
{\it means}, and may call ${\sc a}$ and ${\sc c}$ the
{\it antecedents}, and ${\sc b}$ and ${\sc d}$ the
{\it consequents\/}; we do not disturb the analogy or non-analogy
by interchanging the means among themselves, or the extremes
among themselves; or by altering equally, in direction and in
degree, the two consequents, or the two antecedents, of the
analogy or of the non-analogy, or the two moments of either pair;
or, finally, by altering oppositely in direction, and equally in
degree, the two extremes, or the two means.  In an analogy, we
may also put, by inversion, extremes for means, and means for
extremes; but if a non-analogy be thus inverted, it must
afterwards be changed in kind, from subsequence to precedence, or
from precedence to subsequence.

\bigbreak

\centerline{\it
Combinations of two different analogies, or non-analogies, of
pairs of moments, with}

\nobreak\vskip 3pt

\centerline{\it
each other.}

\nobreak\bigskip

4.
From the remarks last made, it is manifest that
$$\left. \multieqalign{
   &\hbox{if}    &{\sc d}  - {\sc c}  &= {\sc b}  - {\sc a},\cr
   &\hbox{and}   &{\sc d}' - {\sc d}  &= {\sc b}' - {\sc b},\cr
   &\hbox{then}  &{\sc d}' - {\sc c}  &= {\sc b}' - {\sc a};\cr}
   \right\}
   \eqno (20.)$$
because the second of these three analogies shews, that in
passing from the first to the third, we have either made no
change, or only altered equally in direction and in degree the
two consequent moments ${\sc b}$ and ${\sc d}$ of the first
analogy.  In like manner,
$$\left. \multieqalign{
   &\hbox{if}    &{\sc d}  - {\sc c}  &= {\sc b}  - {\sc a},\cr
   &\hbox{and}   &{\sc c}' - {\sc c}  &= {\sc a}' - {\sc a},\cr
   &\hbox{then}  &{\sc d}  - {\sc c}' &= {\sc b}  - {\sc a}';\cr}
   \right\}
   \eqno (21.)$$
because now, in passing from the first to the third analogy, the
second analogy shews that we have either made no change, or else
have only altered equally, in direction and degree, the
antecedents ${\sc a}$ and ${\sc c}$.  Again,
$$\left. \multieqalign{
   &\hbox{if}    &{\sc d}  - {\sc c}  &= {\sc b}  - {\sc a},\cr
   &\hbox{and}   &{\sc d}' - {\sc d}  &= {\sc c}' - {\sc c},\cr
   &\hbox{then}  &{\sc d}' - {\sc c}' &= {\sc b}  - {\sc a};\cr}
   \right\}
   \eqno (22.)$$
because here we have only altered equally, if at all, the two
moments ${\sc c}$ and ${\sc d}$ of one common pair, in passing
from the first analogy to the third.  Again,
$$\left. \multieqalign{
   &\hbox{if}    &{\sc d}  - {\sc c}  &= {\sc b}  - {\sc a},\cr
   &\hbox{and}   &{\sc c}  - {\sc c}' &= {\sc b}' - {\sc b},\cr
   &\hbox{then}  &{\sc d}  - {\sc c}' &= {\sc b}' - {\sc a};\cr}
   \right\}
   \eqno (23.)$$
because now we either do not alter the means ${\sc b}$ and
${\sc c}$ at all, or else alter them oppositely in direction and
equally in degree.  And similarly,
$$\left. \multieqalign{
   &\hbox{if}    &{\sc d}  - {\sc c}  &= {\sc b}  - {\sc a},\cr
   &\hbox{and}   &{\sc d}' - {\sc d}  &= {\sc a}  - {\sc a}',\cr
   &\hbox{then}  &{\sc d}' - {\sc c}  &= {\sc b}  - {\sc a}',\cr}
   \right\}
   \eqno (24.)$$
because here we only alter equally, if at all, in degree, and
oppositely in direction, the extremes, ${\sc a}$ and ${\sc d}$,
of the first analogy.  It is still more evident that if two pairs
be analogous to the same third pair, they are analogous to each
other; that is
$$\left. \multieqalign{
   &\hbox{if}    &{\sc d} - {\sc c}  &= {\sc b}  - {\sc a},\cr
   &\hbox{and}   &{\sc b}  - {\sc a}  &= {\sc d}' - {\sc c}',\cr
   &\hbox{then}  &{\sc d} - {\sc c}  &= {\sc d}' - {\sc c}'.\cr}
   \right\}
   \eqno (25.)$$
And each of the foregoing conclusions will still be true, if we
change the first supposed analogy
${\sc d} - {\sc c} = {\sc b} - {\sc a}$,
to a non-analogy of subsequence
${\sc d} - {\sc c} > {\sc b} - {\sc a}$,
or to a non-analogy of precedence
${\sc d} - {\sc c} < {\sc b} - {\sc a}$,
provided that we change, in like manner, the last or concluded
analogy to a non-analogy of subsequence in the one case, or of
precedence in the other.

It is easy also to see, that if we still suppose the first
analogy
${\sc d} - {\sc c} = {\sc b} - {\sc a}$
to remain, we cannot conclude the third analogy, and are not even
at liberty to suppose that it exists, in any one of the foregoing
combinations, unless we suppose the second also to remain: that
is, if two analogies have the same antecedents, they must have
analogous consequents; if the consequents be the same in two
analogies, the antecedents must themselves form two analogous
pairs; if the extremes of one analogy be the same with the
extremes of another, the means of either may be combined as
extremes with the means of the other as means, to form a new
analogy; if the means of one analogy be the same with the means
of another, then the extremes of either may be combined as means
with the extremes of the other as extremes, and the resulting
analogy will be true; from which the principle of inversion
enables us farther to infer, that if the extremes of one analogy
be the same with the means of another, then the means of the
former may be combined as means with the extremes of the latter
as extremes, and will thus generate another true analogy.

\bigbreak

\centerline{\it
On continued Analogies, or Equidistant Series of Moments.}

\nobreak\bigskip

5.
It is clear from the foregoing remarks, that in any analogy
$${\sc b}' - {\sc a}' = {\sc b} - {\sc a},
   \eqno (26.)$$
the two moments of either pair ${\sc a}$~${\sc b}$ or
${\sc a}'$~${\sc b}'$ cannot coincide, and so reduce themselves
to one single moment, without the two moments of the other pair
${\sc a}'$~${\sc b}'$ or ${\sc a}$~${\sc b}$ being also identical
with each other; not can the two antecedents ${\sc a}$~${\sc a}'$
coincide, without the two consequents ${\sc b}$~${\sc b}'$
coinciding also, nor can the consequents without the antecedents.
The only way, therefore, in which two of the four moments
${\sc a}$~${\sc b}$~${\sc a}'$~${\sc b}'$
of an analogy can coincide, without the two others coinciding
also, that is, the only way in which an analogy can be
constructed with three distinct moments of time, is either by the
two extremes ${\sc a}$~${\sc b}'$ coinciding, or else by the two
means ${\sc b}$~${\sc a}'$ coinciding; and the principle of
inversion permits us to reduce the former of these two cases to
the latter.  We may then take as a sufficient type of every
analogy which can be constructed with three distinct moments, the
following:
$${\sc b}' - {\sc b} = {\sc b} - {\sc a};
   \eqno (27.)$$
that is, the case whan an extreme moment ${\sc b}'$ is related to
a mean moment~${\sc b}$, as that mean moment ${\sc b}$ is related
to another extreme moment~${\sc a}$; in which case we shall say
that the three moments ${\sc a}$~${\sc b}$~${\sc b}'$ compose a
{\it continued analogy}.  In such an analogy, it is manifest that
the three moments ${\sc a}$~${\sc b}$~${\sc b}'$ compose also an
{\it equidistant series}, ${\sc b}'$ being exactly so much later
or so much earlier than ${\sc b}$, as ${\sc b}$ is later or
earlier than ${\sc a}$.  The moment~${\sc b}$ is evidently, in
this case, exactly intermediate between the two other moments
${\sc a}$ and ${\sc b}'$, and may be therefore called {\it the
middle moment}, or the {\it bisector}, of the interval of time
between them.  It is clear that whatever two distinct moments
${\sc a}$ and ${\sc b}'$ may be, there is always one and only one
such bisector moment~${\sc b}$; and that thus a continued analogy
between three moments can always be constructed in one but in
only one way, by inserting a mean, when the extremes are given.
And it is still more evident, from what was shewn before, that
the middle moment~${\sc b}$, along with either of the extremes,
determines the other extreme, so that it is always possible to
complete the analogy in one but in only one way, when an extreme
and the middle are given.

\bigbreak

6.
If, besides the continued analogy (27.) between the three moments
${\sc a}$~${\sc b}$~${\sc b}'$, we have also a continued analogy
between the two last ${\sc b}$~${\sc b}'$ of these three and a
fourth moment ${\sc b}''$, then the {\it four\/} moments
${\sc a}$~${\sc b}$~${\sc b}'$~${\sc b}''$
may themselves also be said to form another {\it continued
analogy}, and an {\it equidistant series}, and we may express
their relations as follows:
$${\sc b}'' - {\sc b}' = {\sc b}' - {\sc b} = {\sc b} - {\sc a}.
   \eqno (28.)$$
In this case, the interval between the two extreme moments
${\sc a}$ and ${\sc b}''$ is {\it trisected\/} by the two
intermediate moments ${\sc b}$ and ${\sc b}'$, and we may call
${\sc b}$ the {\it first trisector}, and ${\sc b}'$ the
{\it second trisector\/} of that interval.  If the first extreme
moment~${\sc a}$ and the first trisector moment~${\sc b}$ be
given, it is evidently possible to complete the continued analogy
or equidistant series in one and in only one way, by supplying
the second trisector~${\sc b}'$ and the second
extreme~${\sc b}''$; and it is not much less easy to perceive
that any two of the four moments being given, (together with
their names of position in the series, as such particular
extremes, or such particular trisectors,) the two other moments
can be determined, as necessarily connected with the given ones.
Thus, if the extremes be given, we must conceive their interval
as being capable of being trisected by two means, in one and in
only one way; if the first extreme and second trisector be given,
we can bisect the interval between them, and so determine (in
thought) the first trisector, and afterwards the second extreme;
if the two trisectors be given, we can continue their interval
equally in opposite directions, and thus determine (in thought)
the two extremes; and if either of these two trisectors along
with the last extreme be given, we can determine, by processes of
the same kind, the two other moments of the series.

\bigbreak

7.
In general, we can imagine a continued analogy and an equidistant
series, comprising any number of moments, and having the interval
between the extreme moments of the series divided into the next
lesser number of portions equal to each other, by a number of
intermediate moments which is itself the next less number to the
number of those equal portions of the whole interval.  For
example, we may imagine an equidistant series of five moments,
with the interval between the two extremes divided into four
partial and mutually equal intervals, by three intermediate
moments, which may be called the first, second, and third
{\it quadrisectors\/} or quarterers of the total interval.  And
it is easy to perceive, that when any two moments of an
equidistant series are given, (as such or such known moments of
time,) together with their places in that series, (as such
particular extremes, or such particular intermediate moments,)
the other moments of the series can then be all determined; and
farther, that the series itself may be continued forward and
backward, so as to include an unlimited number of new moments,
without losing its character of equidistance.  Thus, if we know
the first extreme moment~${\sc a}$, and the third
quadrisector~${\sc b}''$ of the total interval (from ${\sc a}$ to
${\sc b}''$) in any equidistant series of five moments,
${\sc a}$~${\sc b}$~${\sc b}'$~${\sc b}''$~${\sc b}'''$,
we can determine by trisection the two first quadrisectors
${\sc b}$ and ${\sc b}'$, and afterwards the last extreme
moment~${\sc b}'''$; and may then continue the series, forward
and backward, so as to embrace other moments ${\sc b}^{\rm IV}$,
${\sc b}^{\rm V}$, \&c., beyond the fifth of those originally
conceived, and others also such as
${\sc e}$,~${\sc e}'$,~${\sc e}''$, \&c., behind the first of the
original five moments, that is, preceding it in the order of
progression of the series; these new moments forming with the old
an equidistant series of moments, (which comprehends as a part of
itself the original series of five,) namely, the following
unlimited series,
$$\ldots \, {\sc e}'' \, {\sc e}' \, {\sc e} \, {\sc a}
      \, {\sc b} \, {\sc b}' \, {\sc b}'' \, {\sc b}'''
      \, {\sc b}^{\rm IV} \, {\sc b}^{\rm V} \ldots,
   \eqno (29.)$$
constructed so as to satisfy the conditions of a continued
analogy,
$$\eqalignno{\ldots \,
     {\sc b}^{\rm V}  - {\sc b}^{\rm IV}
 &=  {\sc b}^{\rm IV} - {\sc b}'''
  =  {\sc b}'''       - {\sc b}''
  =  {\sc b}''        - {\sc b}' \cr
 &=  {\sc b}'         - {\sc b}
  =  {\sc b}          - {\sc a}
  =  {\sc a}          - {\sc e}
  =  {\sc e} - {\sc e}'
  =  {\sc e}'         - {\sc e}'' \ldots \, .
   &(30.)\cr}$$

\bigbreak

8.
By thus constructing and continuing an equidistant series, of
which any two moments are given, we can arrive at other moments,
as far from those two, and as near to each other, as we desire.
For no moment~${\sc b}$ can be so distant from a given
moment~${\sc a}$, (on either side of it, whether as later or as
earlier,) that we cannot find others still more distant, (and on
the same side of ${\sc a}$, still later or still earlier,) by
continuing (in both directions) any given analogy, or given
equidistant series; and, therefore, no two given moments,
${\sc c}$ and ${\sc d}$, if not entirely coincident, can possibly
be so near to each other, that we cannot find two moments still
more near by treating any two given distinct moments (${\sc a}$
and ${\sc b}$), whatever, as extremes of an equidistant series of
moments sufficiently many, and by inserting the appropriate
means, or intermediate moments, between those two given extremes.
Since, however far it may be necessary to continue the
equidistant series ${\sc c}$~${\sc d} \, \ldots$~${\sc d}''$,
with ${\sc c}$ and ${\sc d}$ for its two first moments, in order
to arrive at a moment~${\sc d}''$ more distant from ${\sc c}$
than ${\sc b}$ is from ${\sc a}$, it is only necessary to insert
as many intermediate moments between ${\sc a}$ and ${\sc b}$ as
between ${\sc c}$ and ${\sc d}''$, in order to generate a new
equidistant series of moments, each nearer to the one next it
than ${\sc d}$ to ${\sc c}$.  Three or more moments
${\sc a}$~${\sc b}$~${\sc c}$ \&c.\ may be said to be
{\it uniserial\/} with each other, when they all belong to one
common continued analogy, or equidistant series; and though we
have not proved (and shall find it not to be true) that
{\it any\/} three moments whatever are thus uniserial moments,
yet we see that if any two moments be given, such as ${\sc a}$
and ${\sc b}$, we can always find a third moment ${\sc b}''$
uniserial with these two, and differing (in either given
direction) by less than any interval proposed from any given
third moment~${\sc c}$, whatever that may be.  This possibility
of indefinitely approaching (on either side) to any given
moment~${\sc c}$, by moments uniserial with any two given ones
${\sc a}$ and ${\sc b}$, increases greatly the importance which
would otherwise belong to the theory of continued analogies, or
equidistant series of moments.  Thus if any two given dates,
${\sc c}$ and ${\sc d}$, denote two distinct moments of time
(${\sc c} \neq {\sc d}$,) however near to each other they may be,
we can always conceive their diversity detected by inserting
means sufficiently numerous between any two other given distinct
moments ${\sc a}$ and ${\sc b}$, as the extremes of an
equidistant series, and then, if necessary, extending this series
in both directions beyond those given extremes, until some one of
the moments~${\sc b}''$ of the equidistant series thus generated
is found to fall between the two mean moments ${\sc c}$ and
${\sc d}$, being later than the earlier, and earlier than the
later of those two.  And, therefore, reciprocally, if in any case
of two given dates ${\sc c}$ and ${\sc d}$, we can prove that no
moment~${\sc b}''$ {\it whatever}, of all that can be imagined as
uniserial with two given distinct moments ${\sc a}$ and
${\sc b}$, falls thus between the moments ${\sc c}$ and
${\sc d}$, we shall then have a sufficient proof that those two
moments ${\sc c}$ and ${\sc d}$ are identical, or, in other
words, that the two dates ${\sc c}$ and ${\sc d}$ represent only
one common moment of time, (${\sc c} = {\sc d}$,) and not two
different moments, however little asunder.

And even in those cases in which we have not yet succeeded in
discovering a rigorous proof of this sort, identifying a sought
moment with a known one, or distinguishing the former from the
latter, the conception of continued analogies offers always a
method of research, and of nomenclature, for investigating and
expressing, or, at least, conceiving as investigated and
expressed, with any proposed degree of approximation if not with
perfect accuracy, the situation of the sought moment in the
general progression of time, by its relation to a known
equidistant series of moments sufficiently close.  This might,
perhaps, be a proper place, in a complete treatise on the
{\it Science of Pure Time}, to introduce a regular system of
{\it integer ordinals}, such as the words {\it first},
{\it second}, {\it third}, \&c., with the written marks
$1$,~$2$,~$3$, \&c., which answer both to them and to the
{\it cardinal\/} or quotitative numbers, {\it one}, {\it two},
{\it three}, \&c.; but it is permitted and required, by the plan
of the present essay, that we should treat these spoken and
written names of the integer ordinals and cardinals, together
with the elementary laws of their combinations, as already known
and familiar.  It is the more admissible in point of method to
suppose this previous acquaintance with the chief properties of
integer numbers, as set forth in elementary arithmetic, because
these properties, although belonging to the Science of Pure Time,
as involving the conception of succession, may all be deduced
from the unfolding of that mere conception of {\it succession},
(among things or thoughts as {\it counted},) without requiring
any notion of {\it measurable intervals}, equal or unequal,
between successive moments of time.  Arithmetic, or the
{\it science of counting}, is, therefore, a part, indeed, of the
{\it Science of Pure Time}, but a part so simple and familiar
that it may be presumed to have been previously and separately
studied, to some extent, by any one who is entering on Algebra.

\bigbreak

\centerline{\it
On steps in the progression of time; their application (direct or
inverse) to moments, so as}

\nobreak\vskip 3pt

\centerline{\it
to generate other moments; and their combination with other
steps, in the way of}

\nobreak\vskip 3pt

\centerline{\it
composition or decomposition.}

\nobreak\bigskip

9.
The foregoing remarks may have sufficiently shewn the importance,
in the general study of pure time, of the conception of a
continued analogy or equidistant series of moments.  This
conception involves and depends on the conception of the repeated
transference of one common ordinal relation, or the continued
application of one common mental step, by which we can pass, in
thought, from any moment of such a series to the moment
immediately following.  For this, and for other reasons, it is
desirable to study, generally, the properties and laws of the
transference, or application, direct or inverse, and of the
composition or decomposition, of ordinal relations between
moments, or of steps in the progression of time; and to form a
convenient system of written signs, for concisely expressing and
reasoning on such applications and such combinations of steps.

In the foregoing articles, we have denoted, by the complex symbol
${\sc b} - {\sc a}$, the ordinal relation of the moment~${\sc b}$
to the moment~${\sc a}$, whether that relation were one of
identity or of diversity; and if of diversity, then whether it
was one of subsequence or of precedence, and in whatever degree.
Thus, having previously interposed the mark~$=$ between two
equivalent signs for one common moment of time, we came to
interpose the same sign of equivalence between any two marks of
one ordinal relation, and to write
$${\sc d} - {\sc c} = {\sc b} - {\sc a},$$
when we designed to express that the relations of ${\sc d}$ to
${\sc c}$ and of ${\sc b}$ to ${\sc a}$ were coincident, being
both relations of identity, or both relations of diversity; and
if the latter, then both relations of subsequence, or both
relations of precedence, and both in the same degree.  In like
manner, having agreed to interpose the mark~$\neq$ between the
two signs of two moments essentially different from each other,
we wrote
$${\sc d} - {\sc c} \neq {\sc b} - {\sc a},$$
when we wished to express that the ordinal relation of ${\sc d}$
to ${\sc c}$ (as identical, or subsequent, or precedent) did
{\it not\/} coincide with the ordinal relation of the
moment~${\sc b}$ to ${\sc a}$; and, more particularly, when we
desired to distinguish between the two principal cases of this
non-coincidence of relations, namely the case when the relation
of ${\sc d}$ to ${\sc c}$ (as compared with that of ${\sc b}$ to
${\sc a}$) was comparatively a relation of lateness, and the case
when the same relation (of ${\sc d}$ to ${\sc c}$) was
comparatively a relation of earliness, we wrote, in the first
case,
$${\sc d} - {\sc c} > {\sc b} - {\sc a},$$
and in the second case,
$${\sc d} - {\sc c} < {\sc b} - {\sc a},$$
having previously agreed to write
$${\sc b} > {\sc a}$$
if the moment ${\sc b}$ were later than the moment~${\sc a}$, or
$${\sc b} < {\sc a}$$
if ${\sc b}$ were earlier than ${\sc a}$.

Now, without yet altering at all the foregoing conception of
${\sc b} - {\sc a}$, as the symbol of an {\it ordinal relation\/}
discovered by the comparison of two moments, we may in some
degree abridge and so far simplify all these foregoing
expressions, by using a simpler symbol of relation, such as a
single letter~${\sr a}$ or ${\sr b}$ \&c.\ or in some cases the
character~$0$, or other simple signs, instead of a complex symbol
such as ${\sc b} - {\sc a}$, or ${\sc d} - {\sc c}$, \&c.  Thus,
if we agree to use the symbol~$0$ to denote the relation of
identity between two moments, writing
$${\sc a} - {\sc a} = 0,
   \eqno (31.)$$
we may express the equivalence of any two dates ${\sc b}$ and
${\sc a}$, by writing
$${\sc b} - {\sc a} = 0,
   \eqno (32.)$$
and may express the non-equivalence of two dates by writing
$${\sc b} - {\sc a} \neq 0;
   \eqno (33.)$$
distinguishing the two cases when the moment~${\sc b}$ is later
and when it is earlier than ${\sc a}$, by writing, in the first
case,
$${\sc b} - {\sc a} > 0,
   \eqno (34.)$$
and in the second case,
$${\sc b} - {\sc a} < 0,
   \eqno (35.)$$
to express, that as compared with the relation of identity~$0$,
the relation ${\sc b} - {\sc a}$ is in the one case a relation of
comparative lateness, and in the other case a relation of
comparative earliness: or, more concisely, by writing, in these
four last cases respectively, which were the cases before marked
(1.) (2.) (3.) and (4.)
$${\sr a} = 0,
   \eqno (36.)$$
$${\sr a} \neq 0,
   \eqno (37.)$$
$${\sr a} > 0,
   \eqno (38.)$$
$${\sr a} < 0,
   \eqno (39.)$$
if we put, for abridgement,
$${\sc b} - {\sc a} = {\sr a}.
   \eqno (40.)$$
Again, if we put, in like manner, for abridgement,
$${\sc d} - {\sc c} = {\sr b},
   \eqno (41.)$$
the analogy (10.) namely,
$${\sc d} - {\sc c} = {\sc b} - {\sc a},$$
may be more concisely expressed as follows,
$${\sr b} = {\sr a};
   \eqno (42.)$$
while the general non-analogy (15.),
$${\sc d} - {\sc c} \neq {\sc b} - {\sc a},$$
may be expressed thus,
$${\sr b} \neq {\sr a},
   \eqno (43.)$$
and the written expressions of its two cases (16.) and (17.),
namely,
$$\eqalign{
{\sc d} - {\sc c} &> {\sc b} - {\sc a} \cr
\hbox{and}\quad
{\sc d} - {\sc c} &< {\sc b} - {\sc a},\cr}$$
may be abridged in the following manner,
$$\eqalignno{
{\sr b} &> {\sr a}, &(44.)\cr
\hbox{and}\quad
{\sr b} &< {\sr a}. &(45.)\cr}$$

Again, to denote a relation which shall be exactly the inverse or
opposite of any proposed ordinal relation ${\sr a}$ or ${\sr b}$,
we may agree to employ a complex symbol such as $\oppos {\sr a}$
or $\oppos {\sr b}$, formed by prefixing the mark~$\oppos$,
(namely, the initial letter~O of the Latin word Oppositio,
distinguished by a bar across it, from the same letter used for
other purposes,) to the mark ${\sr a}$ or ${\sr b}$ of the
proposed ordinal relation; that is, we may agree to use
$\oppos {\sr a}$ to denote the ordinal relation of the
moment~${\sc a}$ to ${\sc b}$, or $\oppos {\sr b}$ to denote the
ordinal relation of ${\sc c}$ to ${\sc d}$, when the
symbol~${\sr a}$ has been already chosen to denote the relation
of ${\sc b}$ to ${\sc a}$, or ${\sr b}$ to denote that of
${\sc d}$ to ${\sc c}$: considering the two assertions
$${\sc b} - {\sc a} = {\sr a},
   \quad\hbox{and}\quad
  {\sc a} - {\sc b} = \oppos {\sr a},
   \eqno (46.)$$
as equivalent each to the other, and in like manner the two
assertions
$${\sc d} - {\sc c} = {\sr b},
   \quad\hbox{and}\quad
  {\sc c} - {\sc d} = \oppos {\sr b},
   \eqno (47.)$$
and similarly in other cases.  In this notation, the theorems
(5.) (6.) (7.) (8.) may be thus respectively written:
$$\oppos {\sr a} = 0,
   \quad\hbox{if}\quad {\sr a} = 0;
   \eqno (48.)$$
$$\oppos {\sr a} \neq 0,
   \quad\hbox{if}\quad {\sr a} \neq 0;
   \eqno (49.)$$
$$\oppos {\sr a} < 0,
   \quad\hbox{if}\quad {\sr a} > 0;
   \eqno (50.)$$
$$\oppos {\sr a} > 0,
   \quad\hbox{if}\quad {\sr a} < 0;
   \eqno (51.)$$
and the theorem of inversion (12.) may be written thus:
$$\oppos {\sr b} = \oppos {\sr a},
   \quad\hbox{if}\quad {\sr b} = {\sr a}.
   \eqno (52.)$$
The corresponding rules for inverting a non-analogy shew that, in
general,
$$\oppos {\sr b} \neq \oppos {\sr a},
   \quad\hbox{if}\quad {\sr b} \neq {\sr a};
   \eqno (53.)$$
and more particularly, that
$$\oppos {\sr b} < \oppos {\sr a},
   \quad\hbox{if}\quad {\sr b} > {\sr a},
   \eqno (54.)$$
and
$$\oppos {\sr b} > \oppos {\sr a},
   \quad\hbox{if}\quad {\sr b} < {\sr a}.
   \eqno (55.)$$
It is evident also that
$$\hbox{if}\quad {\sr a}' = \oppos {\sr a},
   \quad\hbox{then}\quad {\sr a} = \oppos {\sr a}';
   \eqno (56.)$$
that is, the opposite of the opposite of any proposed
relation~${\sr a}$ is that proposed relation itself; a theorem
which may be concisely expressed as follows:
$$\oppos ( \oppos {\sr a} ) = {\sr a};
   \eqno (57.)$$
for, as a general rule of notation, when a complex symbol (as
here $\oppos {\sr a}$) is substituted in any written sentence
(such as here the sentence ${\sr a} = \oppos {\sr a}'$) instead
of a symbol (which the symbol~${\sr a}'$, notwithstanding its
accent, is here considered to be), it is expedient, and in most
cases necessary, for distinctness, to record and mark this using
of a complex as a simple symbol, by some such written warning as
the enclosing of the complex symbol in parentheses, or in
brackets, or the drawing of a bar across it.  However, in the
present case, no confusion would be likely to ensue from the
omission of such a warning; and we might write at pleasure
$$\oppos ( \oppos {\sr a} ) = {\sr a},\quad
  \oppos \{ \oppos {\sr a} \} = {\sr a},\quad
  \oppos [ \oppos {\sr a} ] = {\sr a},\quad
  \oppos \overline{ \oppos {\sr a} } = {\sr a},
   \quad\hbox{or simply}\quad
  \oppos \oppos {\sr a} = {\sr a}.
   \eqno (58.)$$

\bigbreak

10.
For the purpose of expressing, in a somewhat similar notation,
the properties of alternations and combinations of analogies, set
forth in the foregoing articles, with some other connected
results, and generally for the illustration and development of
the conception of ordinal {\it relations\/} between moments, it
is advantageous to introduce that other connected conception,
already alluded to, of {\it steps\/} in the progression of time;
and to establish this other symbolic definition, or conventional
manner of writing, namely,
$${\sc b} = ({\sc b} - {\sc a}) + {\sc a},
   \quad\hbox{or}\quad
  {\sc b} = {\sr a} + {\sc a}
   \quad\hbox{if}\quad
  {\sc b} - {\sc a} = {\sr a};
   \eqno (59.)$$
this notation ${\sr a} + {\sc a}$, or
$({\sc b} - {\sc a}) + {\sc a}$,
corresponding to the above-mentioned conception of a certain
{\it mental step\/} or {\it act of transition}, which is
determined in direction and degree by the ordinal relation
${\sr a}$ or ${\sc b} - {\sc a}$, and may, therefore, be called
``the step~${\sr a}$,'' or the step ${\sc b} - {\sc a}$, and which
is such that by making this mental step, or performing this act
of transition, we pass, in thought, from the moment ${\sc a}$ to
the moment~${\sc b}$, and thus suggest or generate (in thought)
the latter from the former, as a mental product or {\it result\/}
${\sc b}$ of the {\it act\/}~${\sr a}$ and of the
{\it object\/}~${\sc a}$.  We may also express the same relation
between ${\sc b}$ and ${\sc a}$ by writing
$${\sc a} = (\oppos {\sr a}) + {\sc b},
   \quad\hbox{or more simply}\quad
  {\sc a} = \oppos {\sr a} + {\sc b},
   \eqno (60.)$$
if we agree to write the sign~$\oppos {\sr a}$ without
parentheses, as if it were a simple or single symbol, because
there is no danger of causing confusion thereby; and if we
observe that the notation ${\sc a}= \oppos {\sr a} + {\sc b}$
corresponds to the conception of another step, or mental act of
transition, $\oppos {\sr a}$, exactly opposite to the former
step~${\sr a}$, and such that by it we may {\it return\/} (in
thought) from the moment~${\sc b}$ to the moment~${\sc a}$, and
thus may generate ${\sc a}$ as a result of the
act~$\oppos {\sr a}$ and of the object~${\sc b}$.  The mark~$+$,
in this sort of notation, is interposed, as a {\it mark of
combination}, between the signs of the {\it act\/} and the
{\it object}, so as to form a complex sign of the {\it result\/};
or, in other words, between the sign of the transition (${\sr a}$
or $\oppos {\sr a}$) and the sign of the moment (${\sc a}$ or
${\sc b}$) {\it from\/} which that transition is made, so as to
express, by a complex sign, (recording the suggestion or
generation of the thought,) that other moment (${\sc b}$ or
${\sc a}$) {\it to\/} which this mental transition conducts.  And
in any transition of this sort, such as that expressed by the
{\it equation\/} ${\sc b} = {\sr a} + {\sc a}$, we may call (as
before) the moment~${\sc a}$, {\it from\/} which we pass, the
{\it antecedent}, and the moment~${\sc b}$, {\it to\/} which we
pass, the {\it consequent}, of the ordinal
{\it relation\/}~${\sr a}$, or ${\sc b} - {\sc a}$, which
suggests and determines the transition.  In the particular case
when this ordinal relation is one of {\it identity},
(${\sr a} = 0$,) the mental transition or {\it act\/} (${\sr a}$
or $0$) makes no change in the {\it object\/} of that act, namely
in the {\it moment\/}~${\sc a}$, but only leads us to
{\it repeat\/} the thought of that antecedent moment~${\sc a}$,
perhaps with a new name~${\sc b}$; in this case, therefore, the
transition may be said to be {\it null}, or a {\it null step}, as
producing no real alteration in the moment from which it is made.
A step {\it not null}, (${\sr a} \neq 0$,) corresponds to a
relation of {\it diversity}, and may be called, by contrast, an
{\it effective\/} step, because it is an act of thought which
really alters its object, namely the moment to which it is
applied.  An effective step~${\sr a}$ must be either a
{\it late-making\/} or an {\it early-making\/} step, according as
the resultant moment ${\sr a} + {\sc a}$ is later or earlier than
${\sc a}$; but even a {\it null\/} step~$0$ may be regarded as
{\it relatively late-making}, when compared with an early-making
step~${\sr a}$, ($0 + {\sc a} > {\sr a} + {\sc a}$, if
${\sr a} < 0$,) or as {\it relatively early-making\/} if compared
with a late-making step~${\sr b}$;
($0 + {\sc a} < {\sr b} + {\sc a}$, if ${\sr b} > 0$;) and, in
like manner, of two unequal early-making steps, the lesser may be
regarded as relatively late-making, while of two unequal
late-making steps the lesser step may be considered as relatively
early-making.  With these conceptions of the {\it relative
effects\/} of any two steps ${\sr a}$ and ${\sr b}$, we may
enunciate in words the non-analogy (44.), (${\sr b} > {\sr a}$,
that is, ${\sr b} + {\sc a} > {\sr a} + {\sc a}$,) by saying that
the step~${\sr b}$ as compared with the step~${\sr a}$ is
{\it relatively late-making\/}; and the opposite non-analogy
(45.), (${\sr b} < {\sr a}$, that is
${\sr b} + {\sc a} < {\sr a} + {\sc a}$,) by saying that the
step~${\sr b}$ as compared with~${\sr a}$ is {\it relatively
early-making}.

\bigskip

11.
After having made any one step~${\sr a}$ from a proposed
moment~${\sc a}$ to a resulting moment represented (as before) by
${\sr a} + {\sc a}$, we may conceive that we next make from this
new moment ${\sr a} + {\sc a}$ a new step~${\sr b}$, and may
denote the new result by the new complex symbol
${\sr b} + ({\sr a} + {\sc a})$;
enclosing in parentheses the sign ${\sr a} + {\sc a}$ of the
{\it object\/} of this new {\it act\/} of mental transition, or
(in other words) the sign of the new antecedent moment, to mark
that it is a complex used as a simple symbol; so that, in this
notation,
$$\hbox{if}\quad {\sc b} - {\sc a} = {\sr a},
   \quad\hbox{and}\quad {\sc c} - {\sc b} = {\sr b},
   \quad\hbox{then}\quad {\sc c} = {\sr b} + ({\sr a} + {\sc a}).
   \eqno (61.)$$
It is evident that the {\it total change\/} or {\it total step},
effective or null, from the first moment~${\sc a}$ to the last
moment~${\sc c}$, in this successive transition from ${\sc a}$ to
${\sc b}$ and from ${\sc b}$ to ${\sc c}$, may be considered as
{\it compounded\/} of the two successive or {\it partial steps\/}
${\sr a}$ and ${\sr b}$, namely the step~${\sr a}$ from ${\sc a}$
to ${\sc b}$, and the step~${\sr b}$ from ${\sc b}$ to ${\sc c}$;
and that the {\it ultimate ordinal relation\/} of ${\sc c}$ to
${\sc a}$ may likewise be considered as {\it compounded\/} of the
two {\it intermediate\/} (or suggesting) ordinal relations
${\sr b}$ and ${\sr a}$, namely the relation~${\sr b}$ of
${\sc c}$ to ${\sc b}$, and the relation~${\sr a}$ of ${\sc b}$
to ${\sc a}$; a composition of steps or of relations which may
conveniently be denoted, by interposing, as a mark of
combination, between the signs of the component steps or of the
component ordinal relations, the same mark~$+$ which was before
employed to combine an act of transition with its object, or an
ordinal relation with its antecedent.  We shall therefore denote
the compound transition from ${\sc a}$ to ${\sc c}$, or the
compound relation of ${\sc c}$ to ${\sc a}$, by the complex
symbol ${\sr b} + {\sr a}$, writing,
$${\sc c} - {\sc a} = {\sr b} + {\sr a},
   \quad\hbox{if}\quad  {\sc b} - {\sc a} = {\sr a}
   \quad\hbox{and}\quad {\sc c} - {\sc b} = {\sr b},
   \eqno (62.)$$
that is,
$${\sr c} = {\sr b} + {\sr a},
   \quad\hbox{if}\quad  {\sc b} = {\sr a} + {\sc a},\quad
                        {\sc c} = {\sr b} + {\sc b},\quad
                        {\sc c} = {\sr c} + {\sc a}.
   \eqno (63.)$$

For example, the case of coincidence between the moments
${\sc a}$ and ${\sc c}$, that is, the case when the resulting
relation of ${\sc c}$ to ${\sc a}$ is the relation of identity,
and when therefore the total or compound transition from
${\sc a}$ to ${\sc c}$ is null, because the two component or
successive steps ${\sr a}$ and ${\sr b}$ have been exactly
opposite to each other, conducts to the relations,
$$\oppos {\sr a} + {\sr a} = 0;\quad
  {\sr b} + \oppos {\sr b} = 0.
   \eqno (64.)$$

In general, the establishment of this new complex mark
${\sr b} + {\sr a}$, for the compound mental transition from
${\sc a}$ through ${\sc b}$ to ${\sc c}$, permits us to regard
the two written assertions or equations
$${\sc c} = ({\sr b} + {\sr a}) + {\sc a}
   \quad\hbox{and}\quad
  {\sc c} = {\sr b} + ({\sr a} + {\sc a}),
   \eqno (65.)$$
as expressing the same thing, or as each involving the other; for
which reason we are at liberty to omit the parentheses, and may
write, more simply, without fear of causing confusion,
$${\sc c} = {\sr b} + {\sr a} + {\sc a},
   \quad\hbox{if}\quad  {\sc c} = {\sr b} + {\sc b},
   \quad\hbox{and}\quad {\sc b} = {\sr a} + {\sc a}:
   \eqno (66.)$$
because the complex symbol ${\sr b} + {\sr a} + {\sc a}$ denotes
only the one determined moment~${\sc c}$, whether it be
interpreted by first applying the step~${\sr a}$ to the
moment~${\rm {\sc a}}$, so as to generate another moment denoted
by the complex mark ${\sr a} + {\sc a}$, and afterwards applying
to this moment the step denoted by ${\sr b}$, or by first
combining the steps ${\sr a}$ and ${\sr b}$ into one compound
step ${\sr b} + {\sr a}$, and afterwards applying this compound
step to the original moment~${\sc a}$.

In like manner, if three successive steps
${\sr a}$~${\sr b}$~${\sr c}$ have conducted successively (in
thought) from ${\sc a}$ to ${\sc b}$, from ${\sc b}$ to
${\sc c}$, and from ${\sc c}$ to ${\sc d}$, and therefore
ultimately and upon the whole from ${\sc a}$ to ${\sc d}$, we may
consider this total transition from ${\sc a}$ to ${\sc d}$ as
compounded of the three steps ${\sr a}$~${\sr b}$~${\sr c}$; we
may also regard the resulting ordinal relation of ${\sc d}$ to
${\sc a}$ as compounded of the three relations
${\sr c}$, ${\sr b}$, ${\sr a}$, namely the relation~${\sr c}$ of
${\sc d}$ to ${\sc c}$, the relation~${\sr b}$ of ${\sc c}$ to
${\sc b}$, and the relation~${\sr a}$ of ${\sc b}$ to ${\sc a}$;
and may denote this compound step or compound relation by the
complex symbol ${\sr c} + {\sr b} + {\sr a}$, and the last
resulting moment~${\sc d}$ by the connected symbol
${\sr c} + {\sr b} + {\sr a} + {\sc a}$; in such a manner that
$$\left. \eqalign{
  {\sc d} - {\sc a} = {\sr c} + {\sr b} + {\sr a},
   \quad\hbox{and}\quad
  {\sc d} = {\sr c} + {\sr b} + {\sr a} + {\sc a},\cr
   \quad\hbox{if}\quad
      {\sc b} - {\sc a} = {\sr a},\quad
      {\sc c} - {\sc b} = {\sr b},\quad
      {\sc d} - {\sc c} = {\sr c}.\cr}
   \right\}
   \eqno (67.)$$
For example,
$$\left. \multieqalign{
{\sr c} + \oppos {\sr a} + {\sr a} &= {\sr c}, &
{\sr c} + {\sr b} + \oppos {\sr b} &= {\sr c}, \cr
\oppos {\sr b} + {\sr b} + {\sr a} &= {\sr a}, &
{\sr c} + \oppos {\sr c} + {\sr a} &= {\sr a}.\cr}
   \right\}
   \eqno (68.)$$

Remarks of the same kind apply to the composition of more
successive steps than three.  And we see that in any complex
symbol suggested by this sort of composition, such as
${\sr c} + {\sr b} + {\sr a} + {\sc a}$, we are at liberty to
enclose any two or more successive component symbols, such as
${\sr c}$ or ${\sr b}$ or ${\sr a}$ or ${\sc a}$, in parentheses,
with their proper combining marks~$+$, and to treat the enclosed
set as if they formed only one single symbol; thus,
$$\left. \eqalign{
{\sr c} + {\sr b} + {\sr a} + {\sc a}
    =  {\sr c} + {\sr b} + ({\sr a} + {\sc a})
   &=  {\sr c} + ({\sr b} + {\sr a}) + {\sc a} \cr
   &=  ({\sr c} + {\sr b} + {\sr a}) + {\sc a},\quad\hbox{\&c.},\cr}
   \right\}
   \eqno (69.)$$
the notation ${\sr c} + ({\sr b} + {\sr a}) + {\sc a}$, for
example, directing us to begin by combining (in thought) the two
steps ${\sr a}$ and ${\sr b}$ into one compound step
${\sr b} + {\sr a}$, and then to apply successively this compound
step and the remaining step~${\sr c}$ to the original
moment~${\sc a}$; while the notation
$({\sr c} + {\sr b} + {\sr a}) + {\sc a}$
suggests a previous composition (in thought) of all the the three
proposed steps ${\sr a}$, ${\sr b}$, ${\sr c}$, into one compound
step ${\sr c} + {\sr b} + {\sr a}$, and then the application of
this one step to the same original moment.  It is clear that all
these different processes must conduct to one common result; and
generally, that as, by the very meaning and conception of a
{\it compound step}, it may be {\it applied to any moment\/} by
applying in their proper order its component steps successively,
so also may these components be {\it compounded\/} successively
{\it with any other step}, as a mode of compounding with that
other step the whole original compound.

We may also consider {\it decomposition\/} as well as composition
of steps, and may propose to deduce either of two components
${\sr a}$ and ${\sr b}$ from the other component and from the
compound ${\sr b} + {\sr a}$.  For this purpose, it appears from
(68.) that we have the relations
$${\sr a} = \oppos {\sr b} + {\sr c},
   \quad\hbox{and}\quad {\sr b} = {\sr c} + \oppos {\sr a},
   \quad\hbox{if}\quad  {\sr c} = {\sr b} + {\sr a};
   \eqno (70.)$$
observing that a problem of decomposition is plainly a
determinate problem, in the sense that if any one component step,
such as here the step denoted by $\oppos {\sr b} + {\sr c}$, or
that denoted by ${\sr c} + \oppos {\sr a}$, has been found to
conduct to a given compound~${\sr c}$, when combined in a given
order with a given component~${\sr b}$ or ${\sr a}$, then no
other component ${\sr a}$ or ${\sr b}$, essentially different
from the one thus found, can conduct by the same process of
composition to the same given compound step.  We see then that
each of the two components ${\sr a}$ and ${\sr b}$ may be deduced
from the other, and from the compound~${\sr c}$, by compounding
with that given compound the opposite of the given component, in
a suitable order of composition, which order itself we shall
shortly find to be indifferent.

Meanwhile it is important to observe, that though we have agreed,
for the sake of conciseness, to omit the parentheses about a
complex symbol of the kind $\oppos {\sr a}$, when combined with
other written signs by the interposed mark~$+$, yet it is in
general necessary, if we would avoid confusion, to retain the
parentheses, or some such connecting mark or marks, for any
complex symbol of a step, when we wish to form, by prefixing the
mark of opposition~$\oppos$, a symbol for the opposite of that
step: for example, the opposite of a compound step
${\sr b} + {\sr a}$ must be denoted in some such manner as
$\oppos ({\sr b} + {\sr a})$, and not merely by writing
$\oppos {\sr b} + {\sr a}$.  Attending to this remark, we may
write
$$\oppos ({\sr b} + {\sr a})
   = \oppos {\sr a} + \oppos {\sr b},
   \eqno (71.)$$
because, in order to destroy or undo the effect of the compound
step ${\sr b} + {\sr a}$, it is sufficient first to apply the
step $\oppos {\sr b}$ which destroys the effect of the last
component step~${\sr b}$, and afterwards to destroy the effect of
the first component step ${\sr a}$ by applying its opposite
$\oppos {\sr a}$, whatever the two steps denoted by ${\sr a}$
and ${\sr b}$ may be.  In like manner,
$$\oppos ({\sr c} + {\sr b} + {\sr a})
   =  \oppos {\sr a} + \oppos {\sr b} + \oppos {\sr c};
   \eqno (72.)$$
and similarly for more steps than three.

\bigbreak

12.
We can now express, in the language of {\it steps}, several other
general theorems, for the most part contained under a different
form in the early articles of this Essay.

Thus, the propositions (20.) and (21.), with their reciprocals,
may be expressed by saying that if equivalent steps be similarly
combined with equivalent steps, whether in the way of composition
or decomposition, they generate equivalent steps; an assertion
which may be written thus:
$$\left. \multieqalign{
  \hbox{if}\quad  {\sr a}' &= {\sr a},
   \quad\hbox{then}\quad &
  {\sr b} + {\sr a}' &= {\sr b}  + {\sr a}, &
  {\sr a}' + {\sr b} &= {\sr a}' + {\sr b}, \cr
  & &
  {\sr b} + \oppos {\sr a}' &= {\sr b}  + \oppos {\sr a}, &
  \oppos {\sr a}' + {\sr b} &= \oppos {\sr a}  + {\sr b}, \cr
  & &
  \oppos {\sr b} + {\sr a}' &= \oppos {\sr b}  + {\sr a}, &
  {\sr a}' + \oppos {\sr b} &= {\sr a}' + \oppos {\sr b},
      \quad\hbox{\&c.}\cr}
   \right\}
   \eqno (73.)$$

The proposition (25.) may be considered as expressing, that if
two steps be equivalent to the same third step, they are also
equivalent to each other; or, that
$$\hbox{if}\quad
   {\sr a}'' = {\sr a}'
   \quad\hbox{and}\quad
   {\sr a}'  = {\sr a},
   \quad\hbox{then}\quad
   {\sr a}'' = {\sr a}.
   \eqno (74.)$$

The theorem of alternation of an analogy (11.) may be included in
the assertion that in the composition of any two steps, the order
of those two components may be changed, without altering the
compound step; or that
$${\sr a} + {\sr b} = {\sr b} + {\sr a}.
   \eqno (75.)$$
For, whatever the four moments
${\sc a}$~${\sc b}$~${\sc c}$~${\sc d}$ may be, which construct
any proposed analogy or non-analogy, we may denote the step from
${\sc a}$ to ${\sc b}$ by a symbol such as ${\sr a}$, and the
step from ${\sc b}$ to ${\sc d}$ by another symbol~${\sr b}$,
denoting also the step from ${\sc a}$ to ${\sc c}$ by ${\sr b}'$,
and that from ${\sc c}$ to ${\sc d}$ by ${\sr a}'$; in such a
manner that
$${\sc b} - {\sc a} = {\sr a},\quad
  {\sc d} - {\sc b} = {\sr b},\quad
  {\sc c} - {\sc a} = {\sr b}',\quad
  {\sc d} - {\sc c} = {\sr a}';
   \eqno (76.)$$
and then the total step from ${\sc a}$ to ${\sc d}$ may be
denoted either by ${\sr b} + {\sr a}$ or by
${\sr a}' + {\sr b}'$, according as we conceive the transition
performed by passing through ${\sc b}$ or through ${\sc c}$; we
have therefore the relation
$${\sr a}' + {\sr b}' = {\sr b} + {\sr a},
   \eqno (77.)$$
which becomes
$${\sr a} + {\sr b}' = {\sr b} + {\sr a},
   \eqno (78.)$$
when we establish the analogy
$${\sc d} - {\sc c} = {\sc b} - {\sc a},
   \quad\hbox{that is,}\quad {\sr a}' = {\sr a};
   \eqno (79.)$$
we see then that if the theorem (75.) be true, we cannot have the
analogy (79.) without having also its alternate analogy, namely
$${\sr b} = {\sr b}',
   \quad\hbox{or}\quad {\sc d} - {\sc b} = {\sc c} - {\sc a}:
   \eqno (80.)$$
because the compound steps ${\sr a} + {\sr b}'$ and
${\sr a} + {\sr b}$, with the common second component~${\sr a}$,
could not be equivalent, if the first components ${\sr b}'$ and
${\sr b}$ were not also equivalent to each other.  The theorem
(75.) includes, therefore, the theorem of alternation.

Reciprocally, from the theorem of alternation considered as
known, we can infer the theorem (75.), namely, the indifference
of the order of any two successive components
${\sr a}$,~${\sr b}$, of a compound step: for, whatever these two
component steps ${\sr a}$ and ${\sr b}$ may be, we can always
apply them successively to any one moment~${\sc a}$ so as to
generate two other moments ${\sc b}$ and ${\sc c}$, and may again
apply the step ${\sr a}$ to ${\sc c}$ so as to generate a fourth
moment~${\sc d}$, the moments thus suggested having the
properties
$${\sc b} = {\sr a} + {\sc a},\quad
  {\sc c} = {\sr b} + {\sc a},\quad
  {\sc d} = {\sr a} + {\sc c},
   \eqno (81.)$$
and being therefore such that
$${\sc d} - {\sc a} = {\sr a} + {\sr b},\quad
  {\sc d} - {\sc c} = {\sr a} = {\sc b} - {\sc a};
   \eqno (82.)$$
by alternation of which last analogy, between the two pairs of
moments ${\sc a}$~${\sc b}$ and ${\sc c}$~${\sc d}$, we find this
other analogy,
$${\sc d} - {\sc b} = {\sc c} - {\sc a} = {\sr b},\quad
  {\sc d} = {\sr b} + {\sc b} = {\sr b} + {\sr a} + {\sc a},
   \eqno (83.)$$
and finally,
$${\sr b} + {\sr a} = {\sc d} - {\sc a} = {\sr a} + {\sr b}.
   \eqno (84.)$$

The propositions (22.) (23.) (24.), respecting certain
combinations of analogies, are included in the same assertion
(75.); which may also, by (71.), be thus expressed,
$${\sr a} + {\sr b}
   = \oppos (\oppos {\sr a} + \oppos {\sr b}),
   \quad\hbox{or,}\quad
  {\sr b} + {\sr a}
   = \oppos (\oppos {\sr b} + \oppos {\sr a});
   \eqno (85.)$$
that is, by saying that it comes to the same thing, whether we
compound any two steps ${\sr a}$ and ${\sr b}$ themselves, or
first compound their opposites $\oppos {\sr a}$,
$\oppos {\sr b}$, into one compound step
$\oppos {\sr b} + \oppos {\sr a}$, and then take the opposite
of this.  Under this form, the theorem of the possibility of
reversing the order of composition may be regarded as evident,
whatever the number of the component steps may be; for example,
in the case of any three component steps
${\sr a}$, ${\sr b}$, ${\sr c}$, we may regard it as evident that
by applying these three steps successively to any
moment~${\sc a}$, and generating thus three moments
${\sc b}$, ${\sc c}$, ${\sc d}$, we generate moments related to
${\sc a}$ as ${\sc a}$ itself is related to those three other
moments ${\sc b}'$, ${\sc c}'$, ${\sc d}'$, which are generated
from it by applying successively, in the same order, the three
respectively opposite steps, $\oppos {\sr a}$, $\oppos {\sr b}$,
$\oppos {\sr c}$; that is, if
$$\left. \multieqalign{
   {\sc b}  &= {\sr a} + {\sc a}, &
   {\sc b}' &= \oppos {\sr a} + {\sc a},\cr
   {\sc c}  &= {\sr b} + {\sc b}, &
   {\sc c}' &= \oppos {\sr b} + {\sc b}',\cr
   {\sc d}  &= {\sr c} + {\sc c}, &
   {\sc d}' &= \oppos {\sr c} + {\sc c}',\cr}
   \right\}
   \eqno (86.)$$
then the sets
${\sc b}' \, {\sc a} \, {\sc b}$,
${\sc c}' \, {\sc a} \, {\sc c}$,
${\sc d}' \, {\sc a} \, {\sc d}$,
containing each three moments, form so many continued analogies
or equidistant series, such that
$$\left. \eqalign{
  {\sc b} - {\sc a} &= {\sc a} - {\sc b}' \cr
  {\sc c} - {\sc a} &= {\sc a} - {\sc c}' \cr
  {\sc d} - {\sc a} &= {\sc a} - {\sc d}' \cr}
   \right\}
   \eqno (87.)$$
and therefore not only
${\sr b} + {\sr a}
   = \oppos ( \oppos {\sr b} + \oppos {\sr a})$,
as before, but also
$${\sr c} + {\sr b} + {\sr a}
   = \oppos ( \oppos {\sr c} + \oppos {\sr b} + \oppos {\sr a}),
   \eqno (88.)$$
that is, by (72.) and (57.),
$${\sr c} + {\sr b} + {\sr a}
   =  {\sr a} + {\sr b} + {\sr c};
   \eqno (89.)$$
and similarly for more steps than three.

The theorem (89.) was contained, indeed, in the reciprocal of the
proposition (24.), namely, in the assertion that
$$\left. \multieqalign{
   &\hbox{if}    &{\sc d} - {\sc c}  &= {\sc b}  - {\sc a},\cr
   &\hbox{and}   &{\sc d}' - {\sc c}  &= {\sc b}  - {\sc a}',\cr
   &\hbox{then}  &{\sc d}' - {\sc d}  &= {\sc a}  - {\sc a}',\cr}
   \right\}
   \eqno (90.)$$
and therefore, by alternation,
$${\sc d}' - {\sc a} = {\sc d} - {\sc a}';
   \eqno (91.)$$
for, whatever the three steps ${\sr a}$~${\sr b}$~${\sr c}$ may
be, we may always conceive them applied successively to any
moment~${\sc a}$, so as to generate three other moments
${\sc b}$,~${\sc c}$,~${\sc d}'$,
such that
$${\sc b}  = {\sr a} + {\sc a},\quad
  {\sc c}  = {\sr b} + {\sc b},\quad
  {\sc d}' = {\sr c} + {\sc c},
   \eqno (92.)$$
and may also conceive two other moments ${\sc a}'$ and ${\sc d}$
such that ${\sc b}$~${\sc c}$~${\sc d}$ may be successively
generated from ${\sc a}'$ by applying the same three steps in the
order ${\sr c}$, ${\sr b}$, ${\sr a}$, so that
$${\sc b} = {\sr c} + {\sc a}',\quad
  {\sc c} = {\sr b} + {\sc b},\quad
  {\sc d} = {\sr a} + {\sc c};
   \eqno (93.)$$
and then the first two analogies of the combination (90.) will
hold, and, therefore, also the last, together with its alternate
(91.); that is, the step from ${\sc a}$ to ${\sc d}'$, compounded
of the three steps ${\sr a}$~${\sr b}$~${\sr c}$, is equivalent
to the step from ${\sc a}'$ to ${\sc d}$, compounded of the same
three steps in the reverse order ${\sr c}$~${\sr b}$~${\sr a}$.

Since we may thus reverse the order of any three successive
steps, and also the order of any two which immediately follow
each other, it is easy to see that we may interchange in any
manner the order of three successive steps; thus
$$\left. \eqalign{
{\sr c} + {\sr b} + {\sr a}
   = {\sr c} + {\sr a} + {\sr b}
   = {\sr b} + {\sr c} + {\sr a} \phantom{.} \cr
   = {\sr a} + {\sr b} + {\sr c}
   = {\sr a} + {\sr c} + {\sr b}
   = {\sr b} + {\sr a} + {\sr c}.\cr}
   \right\}
   \eqno (94.)$$

We might also have proved this theorem (94.), without previously
establishing the less general proposition (89.), and in a manner
which would extend to any number of component steps; namely, by
observing that when any arrangement of component steps is
proposed, we may always reserve the first (and by still stronger
reason any other) of those steps to be applied the last, and
leave the order of the remaining steps unchanged, without
altering the whole compound step; because the components which
followed, in the proposed arrangement, that one which we now
reserve for the last, may be conceived as themselves previously
combined into one compound step, and this then interchanged in
place with the reserved one, by the theorem respecting the
arbitrary order of any two successive steps.  In like manner, we
might reserve any other step to be the last but one, and any
other to be last but two, and so on; by pursuing which reasoning
it becomes manifest that when any number of component steps are
applied to any original moment, or compounded with any primary
step, their order may be altered at pleasure, without altering
the resultant moment, or the whole compounded step: which is,
perhaps, the most important and extensive property of the
composition of ordinal relations, or steps in the progression of
time.

\bigbreak

\centerline{\it
On the Multiples of a given base, or unit-step; and on the
Algebraic Addition,}

\nobreak\vskip 3pt

\centerline{\it
Subtraction, Multiplication, and Division, of
their determining or}

\nobreak\vskip 3pt

\centerline{\it
multipling Whole Numbers, whether positive, or contra-positive, or null.}

\nobreak\bigskip

13.
Let us now apply this general theory of successive and compound
steps, from any one moment to any others, or of component and
compound ordinal relations between the moments of any arbitrary
set, to the case of an equidistant series of moments,
$$\ldots \, {\sc e}'' \, {\sc e}' \, {\sc e}
   \, {\sc a} \, {\sc b} \, {\sc b}' \, {\sc b}'' \ldots
   \eqno (29.)$$
constructed so as to satisfy the conditions of a continued
analogy,
$$\ldots \,
      {\sc b}'' - {\sc b}'
   =  {\sc b}'  - {\sc b}
   =  {\sc b}   - {\sc a}
   =  {\sc a}   - {\sc e}
   =  {\sc e} - {\sc e}'
   =  {\sc e}'  - {\sc e}'',
   \quad\hbox{\&c.};
   \eqno (30.)$$
and first, for distinctness of conception and of language, let
some one moment~${\sc a}$ of this series be selected as a
standard with which all the others are to be compared, and let it
be called the {\it zero-moment\/}; while the moments
${\sc b}$,~${\sc b}'$ \&c.\ which {\it follow\/} it, in the order
of progression of the series, may be distinguished from those
other moments ${\sc e}$,~${\sc e}'$, \&c., which {\it precede\/}
it in that order of progression, by some two contrasted epithets,
such as the words {\it positive\/} and {\it contra-positive\/}:
the moment~${\sc b}$ being called the {\it positive first}, or
the first moment of the series on the positive side of the zero;
while in the same plan of nomenclature the moment~${\sc b}'$ is
the {\it positive second}, ${\sc b}''$ is the {\it positive
third}, ${\sc e}$ the {\it contra-positive first}, ${\sc e}'$ the
{\it contra-positive second}, and so forth.  By the nature of the
series, as composed of equi-distant moments, or by the conditions
(30.), all the positive or {\it succeeding\/} moments
${\sc b}$~${\sc b}'$ \&c.\ may be conceived as {\it generated\/}
from the zero-moment~${\sc a}$, by the continual and successive
application of one common step~${\sr a}$, and all the
contra-positive or {\it preceding\/} moments
${\sc e}$,~${\sc e}'$~\&c.\ may be conceived as generated from
the same zero-moment~${\sc a}$, by the continual and successive
application of the opposite step $\oppos {\sr a}$, so that we may
write
$${\sc b} =   {\sr a} + {\sc a},\quad
  {\sc b}' =  {\sr a} + {\sc b},\quad
  {\sc b}'' = {\sr a} + {\sc b}',\quad
   \hbox{\&c.},
   \eqno (95.)$$
and
$${\sc e} =   \oppos {\sr a} + {\sc a},\quad
  {\sc e}' =  \oppos {\sr a} + {\sc e},\quad
  {\sc e}'' = \oppos {\sr a} + {\sc e}';\quad
   \hbox{\&c.};
   \eqno (96.)$$
while the standard or zero-moment~${\sc a}$ itself may be denoted
by the complex symbol $0 + {\sc a}$, because it may be conceived
as generated from itself by applying the null step~$0$.  Hence,
by the theory of compound steps, we have expressions of the
following sort for all the several moments of the equidistant
series (29.):
$$\left. \eqalign{
\noalign{\vbox{\hbox{$\cdots\cdots$}}}
{\sc e}'' &= \oppos {\sr a} + \oppos {\sr a} + \oppos {\sr a} + {\sc a},\cr
{\sc e}'  &= \oppos {\sr a} + \oppos {\sr a} + {\sc a},\cr
{\sc e}   &= \oppos {\sr a} + {\sc a},\cr
{\sc a}   &= 0 + {\sc a},\cr
{\sc b}   &= {\sr a} + {\sc a},\cr
{\sc b}'  &= {\sr a} + {\sr a} + {\sc a},\cr
{\sc b}'' &= {\sr a} + {\sr a} + {\sr a} + {\sc a},\cr
\noalign{\vbox{\hbox{$\cdots\cdots$}}}}
   \right\}
   \eqno (97.)$$
with corresponding expressions for their several ordinal
relations to the one standard moment~${\sc a}$, or for the acts
of transition which are made in passing from ${\sc a}$ to them,
namely:
$$\left. \eqalign{
\noalign{\vbox{\hbox{$\cdots\cdots$}}}
{\sc e}'' - {\sc a} &= \oppos {\sr a} + \oppos {\sr a} + \oppos {\sr a},\cr
{\sc e}'  - {\sc a} &= \oppos {\sr a} + \oppos {\sr a},\cr
{\sc e}   - {\sc a} &= \oppos {\sr a},\cr
{\sc a}   - {\sc a} &= 0,\cr
{\sc b}   - {\sc a} &= {\sr a},\cr
{\sc b}'  - {\sc a} &= {\sr a} + {\sr a},\cr
{\sc b}'' - {\sc a} &= {\sr a} + {\sr a} + {\sr a},\cr
\hbox{\&c.} \cr}
   \right\}
   \eqno (98.)$$

The simple or compound step, ${\sr a}$, or ${\sr a} + {\sr a}$,
\&c., from the zero moment~${\sc a}$ to any positive moment
${\sc b}$ or ${\sc b}'$ \&c.\ of the series, may be called a
{\it positive step\/}; and the opposite simple or compound step
$\oppos {\sr a}$, or $\oppos {\sr a} + \oppos {\sr a}$, \&c.,
from the same zero-moment~${\sc a}$ to any contra-positive
moment~${\sc e}$ or ${\sc e}'$, \&c., of the series, may be
called a {\it contra-positive step\/}; while the null step~$0$,
from the zero-moment ${\sc a}$ to itself, may be called, by
analogy of language, the {\it zero-step}.  The original
step~${\sr a}$ is supposed to be an effective step, and not a
null one, since otherwise the whole series of moments (97.) would
reduce themselves to the one original moment~${\sc a}$; but it
may be either a late-making or an early-making step, according as
the (mental) order of progression of that series is from earlier
to later, or from later to earlier moments.  And the whole series
or system of steps (98.), simple or compound, positive or
contra-positive, effective or null, which serve to generate the
several moments of the equi-distant series (29.) or (97.) from
the original or standard moment~${\sc a}$, may be regarded as a
{\it system of steps generated from the original
step\/}~${\sr a}$, by a {\it system of acts\/} of generation
which are all of one common kind; each step having therefore a
certain {\it relation\/} of its own to that original step, and
these relations having all a general resemblance to each other,
so that they may be conceived as composing a certain {\it system
of relations}, having all one common character.  To mark this
{\it common generation\/} of the system of steps (98.) from the
one original step~${\sr a}$, and their {\it common relation\/}
thereto, we may call them all by the common name of
{\it multiples\/} of that original step, and may say that they
are or may be (mentally) formed by {\it multipling\/} that
common {\it base}, or {\it unit-step},~${\sr a}$; distinguishing,
however, these several multiples among themselves by peculiar or
special names, which shall serve to mark the peculiar relation of
any one multiple to the base, or the special act of multipling
by which it may be conceived to be generated therefrom.

Thus the null step, or zero-step, $0$, which conducts to the
zero-moment~${\sc a}$, may be called, according to this way of
conceiving it, the {\it zero multiple\/} of the original
step~${\sr a}$; and the positive (effective) steps, simple or
compound, ${\sr a}$, ${\sr a} + {\sr a}$,
${\sr a} + {\sr a} + {\sr a}$, \&c., may be called by the general
name of {\it positive multiples\/} of ${\sr a}$, and may be
distinguished by the special ordinal names of {\it first},
{\it second}, {\it third}, \&c., so that the original
step~${\sr a}$ is, in this view, its own first positive multiple;
and finally, the contra-positive (but effective) steps, simple or
compound, namely, $\oppos {\sr a}$,
$\oppos {\sr a} + \oppos {\sr a}$,
$\oppos {\sr a} + \oppos {\sr a} + \oppos {\sr a}$, \&c,
may be called the {\it first contra-positive multiple\/} of
${\sr a}$, the {\it second contra-positive multiple\/} of the
same original step~${\sr a}$, and so forth.  Some particular
multiples have particular and familiar names; for example, the
second positive multiple of a step may also be called the
{\it double\/} of that step, and the third positive multiple may
be called familiarly the {\it triple}.  In general, the original
step~${\sr a}$ may be called (as we just now agreed) the common
{\it base\/} ({\it or unit\/}) of all these several multiples;
and the ordinal name or number, (such as zero, or positive first,
or contra-positive second,) which serves as a special mark to
distinguish some one of these multiples from every other, in the
general series of such multiples (98.), may be called the
{\it determining ordinal\/}: so that any one multiple step is
sufficiently described, when we mention its base and its
determining ordinal.  In conformity with this conception of the
series of steps (98.), as a {\it series of multiples of the
base\/}~${\sr a}$, we may denote them by the following series of
written symbols,
$$\ldots \, 3 \oppos {\sr a},\quad 2 \oppos {\sr a},\quad
   1 \oppos {\sr a},\quad 0 {\sr a},\quad 1 {\sr a},\quad
   2 {\sr a},\quad 3 {\sr a},\ldots
   \eqno (99.)$$
and may denote the moments themselves of the equi-distant series
(29.) or (97.) by the symbols,
$$\left. \eqalign{
\noalign{\vbox{\hbox{$\cdots\cdots$}}}
{\sc e}'' &= 3 \oppos {\sr a} + {\sc a},\cr
{\sc e}'  &= 2 \oppos {\sr a} + {\sc a},\cr
{\sc e}   &= 1 \oppos {\sr a} + {\sc a},\cr
{\sc a}   &= 0 {\sr a} + {\sc a},\cr
{\sc b}   &= 1 {\sr a} + {\sc a},\cr
{\sc b}'  &= 2 {\sr a} + {\sc a},\cr
{\sc b}'' &= 3 {\sr a} + {\sc a},\cr
\hbox{\&c.;} \cr}
   \right\}
   \eqno (100.)$$
in which
$$0 {\sr a} = 0,
   \eqno (101.)$$
and
$$\left. \multieqalign{
1 {\sr a} &= {\sr a}, &
1 \oppos {\sr a} &= \oppos {\sr a},\cr
2 {\sr a} &= {\sr a} + {\sr a}, &
2 \oppos {\sr a} &= \oppos {\sr a} + \oppos {\sr a},\cr
3 {\sr a} &= {\sr a} + {\sr a} + {\sr a}, &
3 \oppos {\sr a} &= \oppos {\sr a} + \oppos {\sr a} + \oppos {\sr a},\cr
   &\mathrel{\phantom{=}}\hbox{\&c.,} &
   &\mathrel{\phantom{=}}\hbox{\&c.} \cr}
   \right\}
   \eqno (102.)$$

The written sign~$0$ in $0 {\sr a}$ is here equivalent to the
spoken name {\it zero}, as the determining ordinal of the null
step from ${\sc a}$ to ${\sc a}$, which step was itself also
denoted before by the same character~$0$, and is here considered
as the {\it zero-multiple\/} of the base~${\sr a}$; while the
written signs $1$,~$2$,~$3$, \&c., in the symbols of the positive
multiples $1 {\sr a}$, $2 {\sr a}$, $3 {\sr a}$, \&c., correspond
to and denote the determining positive ordinals, or the spoken
names {\it first positive}, {\it second positive}, {\it third
positive}, \&c.; and, finally, the remaining written signs
$1 \oppos$, $2 \oppos$, $3 \oppos$, \&c., which are combined with
the written sign of the base~${\sr a}$, in the symbols of the
contra-positive multiples $1 \oppos {\sr a}$, $2 \oppos {\sr a}$,
$3 \oppos {\sr a}$, \&c., correspond to and denote the
determining ordinal names of those contra-positive multiples,
that is, they correspond to the spoken names, {\it first
contra-positive}, {\it second contra-positive},
{\it third contra-positive}, \&c.: so that the series of signs of
multiple steps (99.), is formed by combining the symbol of the
base~${\sr a}$ with the following series of ordinal symbols,
$$\ldots, 3 \oppos,\quad 2 \oppos,\quad 1 \oppos,\quad
  0,\quad 1,\quad 2,\quad 3,\quad \hbox{\&c.}
   \eqno (103.)$$

We may also conceive this last series of signs as equivalent, not
to {\it ordinal\/} names, such as the numeral word {\it first},
but to {\it cardinal\/} names, such as the numeral word
{\it one\/}; or more fully, {\it positive cardinals},
{\it contra-positive cardinals}, and the {\it null cardinal\/}
(or number {\it none\/}); namely, the system of all possible
answers to the following complex question: ``{\it Have any\/}
effective steps (equivalent or opposite to the given
base~${\sr a}$) been made (from the standard moment~${\sc a}$),
and if any, then {\it How many}, and {\it In which
direction\/}?'' In this view, $3 \oppos$ is a written sign of the
{\it cardinal\/} name or number {\it contra-positive three}, as a
possible answer to the foregoing general question; and it
implies, when prefixed to the sign of the base~${\sr a}$, in the
complex written sign $3 \oppos {\sr a}$ of the corresponding
multiple step, that this multiple step has been formed, (as
already shown in the equations (102.),) by making three steps
equal to the base~${\sr a}$ in length, but in the direction
opposite thereto.  Again, the mark~$1$ may be regarded as a
written sign of the cardinal number {\it positive one}, and $1$
denotes (in this view) the step formed by making one such step as
${\sr a}$, and in the same direction, that is, (as before,) the
original step ${\sr a}$ itself; and $0$ denotes the cardinal
number {\it none}, so that $0 {\sr a}$ is (as before) a symbol
for the null step from ${\sc a}$ to ${\sc a}$, which step we have
also marked before by the simple symbol~$0$, and which is here
considered as formed by making {\it no\/} effective step like
${\sr a}$.  In general, this view of the numeral signs (103.), as
denoting {\it cardinal\/} numbers, conducts to the same ultimate
interpretations of the symbols (99.), for the steps of the series
(98.), as the former view, which regarded those signs (103.) as
denoting {\it ordinal\/} numbers.

If we adopt the latter view of those numeral signs (103.), which
we shall call by the common name of {\it whole\/} (or
{\it integer\/}) {\it numbers}, (as distinguished from certain
broken or fractional numbers to be considered afterwards,) we may
conveniently continue to use the word {\it multiple\/}
(occasionally) as a verb active, and may speak of the several
multiple steps of the series (98.), or (99.), as formed from the
base~${\sr a}$, by {\it multipling that base by the several
whole\/} (cardinal) {\it numbers\/}: because every multiple step
may be conceived as generated (in thought) from the base, by a
certain mental act, of which the cardinal number is the mark.
Thus we may describe the multiple step $3 \oppos {\sr a}$,
(which is, in the ordinal view, the third contra-positive
multiple of ${\sr a}$,) as formed from the base~${\sr a}$ by
{\it multipling it by contra-positive three}.  Some particular
acts of multipling have familiar and special names, and we may
speak (for instance) of {\it doubling\/} or {\it tripling\/} a
step, instead of describing that step as being multipled by
positive two, or by positive three.  In general, to distinguish
more clearly, in the written symbol of a multiple step, between
the base and the determining number (ordinal or cardinal), and to
indicate more fully the performance of that mental act (directed
by the number) which generates the multiple from the base, the
mark~$\times$ may be inserted between the sign of the base, and
the sign of the number; and thus we may denote the series of
multiple steps (99.) by the following fuller symbols,
$$\ldots
      3 \oppos \times {\sr a},\quad
      2 \oppos \times {\sr a},\quad
      1 \oppos \times {\sr a},\quad
      0 \times {\sr a},\quad
      1 \times {\sr a},\quad
      2 \times {\sr a},\quad
      3 \times {\sr a},\quad \hbox{\&c.},
   \eqno (104.)$$
and which $1 \times {\sr a}$ (for example) denotes the original
step~${\sr a}$ itself, and $2 \times {\sr a}$ represents the
double of that original step.

It is manifest that in this notation
$$\left. \eqalign{
n \oppos \times {\sr a}
   = n \times \oppos {\sr a}
   = \oppos ( n \times {\sr a} )
   = \oppos ( n \oppos \times \oppos {\sr a} ),\cr
\hbox{and}\quad
n \times {\sr a}
   = n \oppos \times \oppos {\sr a}
   = \oppos (n \oppos \times {\sr a})
   = \oppos (n \times \oppos {\sr a}),\cr}
   \right\}
   \eqno (105.)$$
if $n$ denote any one of the positive numbers $1$,~$2$,~$3$,
\&c.\ and if $n \oppos$ denote the corresponding contra-positive
number, $1 \oppos$, $2 \oppos$, $3 \oppos$, \&c.; for example,
the equation
$2 \oppos \times {\sr a} = 2 \times \oppos {\sr a}$
is true, because it expresses that the second contra-positive
multiple of the base~${\sr a}$ is the same step as the second
positive multiple of the opposite base or step $\oppos {\sr a}$,
the latter multiple being derived from this opposite base by
merely doubling its length without reversing its direction, while
the former is derived from the original base~${\sr a}$ itself by
both reversing it in direction and doubling it in length, so that
both processes conduct to the one common compound step,
$\oppos {\sr a} + \oppos {\sr a}$.
In like manner the equation
$2 \times {\sr a} = 2 \oppos \times \oppos {\sr a}$
is true, because by first reversing the direction of the original
step~${\sr a}$, and then taking the reversed step
$\oppos {\sr a}$ as a new base, and forming the second
contra-positive multiple of it, which is done by reversing and
doubling, and which is the process of generation expressed by the
symbol $2 \oppos \times \oppos {\sr a}$, we form in the end the
same compound step, ${\sr a} + {\sr a}$, as if we had merely
doubled ${\sr a}$.  We may also conveniently annex the mark of
opposition~$\oppos$, at the left hand, to the symbol of any
whole number,~$n$ or $n \oppos$ or $0$, in order to form a
symbol of its opposite number $n \oppos$, $n$, or $0$; and thus
may write
$$\oppos n = n \oppos,\quad
  \oppos (n \oppos) = n,\quad
  \oppos 0 = 0;
   \eqno (106.)$$
if we still denote by $n$ any positive whole number, and if we
call two whole numbers {\it opposites\/} of each other, when they
are the determining or multipling numbers of two opposite
multiple steps.

\bigbreak

14.
Two or more multiples such as
$\mu \times {\sr a}$, $\nu \times {\sr a}$, $\xi \times {\sr a}$,
of the same base~${\sr a}$, may be compounded as {\it successive
steps\/} with each other, and the resulting or compound step will
manifestly be itself some multiple, such as
$\omega \times {\sr a}$, of the same common base~${\sr a}$; the
signs $\mu$, $\nu$, $\xi$, denoting here any arbitrary whole
numbers, whether positive, or contra-positive, or null, and
$\omega$ denoting another whole number, namely the determining
number of the compound multiple step, which must evidently depend
on the determining numbers $\mu$~$\nu$~$\xi$ of the component
multiple steps, and on those alone, according to some general law
of dependence.  This law may conveniently be denoted, in writing,
by the same mark of combination~$+$ which has been employed
already to form the complex symbol of the compound step itself,
considered as depending on the component steps; that is, we may
agree to write
$$\omega = \nu + \mu,
   \quad\hbox{when}\quad
  \omega \times {\sr a}
      = (\nu \times {\sr a}) + (\mu \times {\sr a}),
   \eqno (107.)$$
and
$$\omega = \xi + \nu + \mu,
   \quad\hbox{when}\quad
  \omega \times {\sr a}
      = (\xi \times {\sr a}) + (\nu \times {\sr a}) + (\mu \times {\sr a}),
   \eqno (108.)$$
together with other similar expressions for the case of more
component steps than three.  In this notation,
$$\left. \eqalign{
(\nu \times {\sr a}) + (\mu \times {\sr a})
   &= (\nu + \mu) \times {\sr a},\cr
(\xi \times {\sr a}) + (\nu \times {\sr a}) + (\mu \times {\sr a})
   &= (\xi + \nu + \mu) \times {\sr a},\cr
   \hbox{\&c.}\cr}
   \right\}
   \eqno (109.)$$
whatever the whole numbers $\mu$~$\nu$~$\xi$ may be; equations
which are to be regarded here as true by definition, and as only
serving to explain the meaning attributed to such complex signs
as $\nu + \mu$, or $\xi + \nu + \mu$, when $\mu$~$\nu$~$\xi$ are
any symbols of whole numbers: although when we farther assert
that the equations (109.) are true independently of the base or
unit-step~${\sr a}$, so that symbols of the form $\nu + \mu$ or
$\xi + \nu + \mu$ denote whole numbers independent of that
base, we express in a new way a theorem which we had before
assumed to be evidently true, as an axiom and not a definition,
respecting the composition of multiple steps.

In the particular case when the whole numbers denoted by
$\mu$~$\nu$~$\xi$ are positive, the law of composition of those
numbers expressed by the notation
$\nu + \mu$ or $\xi + \nu + \mu$, as explained by the equations
(109.), is easily seen to be the law called {\it addition\/} of
numbers (that is of quotities) in elementary arithmetic; and the
quotity of the compound or resulting whole number is the
arithmetical {\it sum\/} of the quotities of the component
numbers, this arithmetical {\it sum\/} being the answer to the
question, {\it How many\/} things or thoughts does a total group
contain, if it be composed of {\it partial groups\/} of which the
quotities are given, namely the numbers to be arithmetically
{\it added}.  For example, since
$(3 + {\sr a}) + (2 \times {\sr a})$
is the symbol for the total or compound multiple step composed of
the double and the triple of the base~${\sr a}$, it must denote
the quintuple or fifth positive multiple of that base, namely
$5 \times {\sr a}$; and since we have agreed to write
$$(3 \times {\sr a}) + (2 \times {\sr a})
  = (3 + 2) \times {\sr a},$$
we must interpret the complex symbol $3 + 2$ as equivalent to the
simple symbol~$5$; in seeking for which latter number {\it five},
we {\it added}, in an arithmetical sense, the given component
numbers {\it two\/} and {\it three\/} together, that is, we
formed their arithmetical {\it sum}, by considering how many
steps are contained in a total group of steps, if the component
or partial groups contain two steps and three steps respectively.
In like manner, if we admit in arithmetic the idea of the
cardinal number {\it none}, as one of the possible answers to the
fundamental question {\it How many}, the rules of the
arithmetical addition of this number to others, and of others to
it, and the properties of the arithmetical sums thus composed,
agree with the rules and properties of such combinations as
$0 + \mu$, $\xi + \nu + 0$, explained by the equations (109.),
when the whole numbers $\mu$, $\nu$, $\xi$ are positive; we
shall, therefore, not clash in our enlarged phraseology with the
language of elementary arithmetic, respecting the addition of
numbers regarded as answers to the question {\it How many}, if we
now establish, as a definition, in the more extensive {\it Scince
of Pure Time}, that any combination of whole numbers
$\mu$~$\nu$~$\xi$, of the form $\nu + \mu$, or $\xi + \nu + \mu$,
interpreted so as to satisfy the equations (109.), is the
{\it sum\/} of those whole numbers, and is composed by
{\it adding\/} them together, whether they be positive, or
contra-positive, or null.  But as a mark that these words
{\it sum\/} and {\it adding\/} are used in {\sc Algebra} (as the
general Science of Pure Time), in a more extensive sense than
that in which {\it Arithmetic\/} (as the science of counting)
employs them, we may, more fully, call $\nu + \mu$ the
{\it algebraic sum\/} of the whole numbers $\mu$ and $\nu$, and
say that it is formed by the operation of {\it algebraically
adding\/} them together, $\nu$ to $\mu$.

In general, we may extend the arithmetical names of {\it sum\/}
and {\it addition\/} to every algebraical combination of the
class marked by the sign~$+$, and may give to that combining sign
the arithmetical name of {\it Plus\/}; although in Algebra the
idea of {\it more}, (originally implied by {\it plus},) is only
occasionally and accidentally involved in the conception of such
combinations.  For example, the written symbol
${\sr b} + {\sr a}$, by which we have already denoted the
compound step formed by {\it compounding\/} the step~${\sr b}$ as
a successive step with the step~${\sr a}$, may be expressed in
words by the phrase ``${\sr a}$ plus ${\sr b}$'' (such written
algebraic expressions as these being read from right to left,) or
``the algebraic sum of the steps ${\sr a}$ and ${\sr b}$;'' and
this algebraic sum or compound step ${\sr b} + {\sr a}$ may be
said to be formed by ``algebraically adding ${\sr b}$ to
${\sr a}$:'' although this compound step is only occasionally and
accidentally greater in length than its components, being
necessarily shorter than one of them, when they are both
effective steps with directions opposite to each other.  Even the
{\it application\/} of a step~${\sr a}$ to a moment~${\sc a}$, so
as to generate another moment ${\sr a} + {\sc a}$, may not
improperly be called (by the same analogy of language) the
{\it algebraic addition\/} of the step to the moment, and the
moment generated thereby may be called their {\it algebraic sum},
or ``the original moment {\it plus\/} the step;'' though in this
sort of combination the moment and the step to be combined are
not even homogeneous with each other.

With respect to the process of calculation of an algebraic sum of
whole numbers, the following rules are evident consequences of
what has been already shown respecting the composition of steps.
In the first place, the numbers to be added may be added in any
arbitrary order; that is,
$$\left. \eqalign{
   &\nu + \mu = \mu + \nu,\cr
   &\xi + \nu + \mu = \mu + \xi + \nu = \hbox{\&c.},\cr
   &\quad\hbox{\&c.};\cr}
  \right\}
   \eqno (110.)$$
we may therefore collect the positive numbers into one
algebraical sum, and the contra-positive into another, and then
add these two partial sums to find the total sum, omitting (if it
anywhere occur) the number None or Zero, as not capable of
altering the result.  In the next place, positive numbers are
algebraically added to each other, by arithmetically adding the
corresponding arithmetical numbers or quotities, and considering
the result as a positive number; thus positive two and positive
three, when added, give positive five: and contra-positive
numbers, in like manner, are algebraically added to each other,
by arithmetically adding their quotities, and considering the
result as a contra-positive number; thus, contra-positive two and
contra-positive three have contra-positive five for their
algebraic sum.  In the third place, a positive number and a
contra-positive, when the quotity of the positive exceeds that of
the contra-positive, give a positive algebraic sum, in which the
quotity is equal to that excess; thus positive five added to
contra-positive three, gives positive two for the algebraic sum:
and similarly, a positive number and a contra-positive number, if
the quotity of the contra-positive exceed that of the positive,
give a contra-positive algebraic sum, with a quotity equal to the
excess; for example, if we add positive three to contra-positive
five, we get contra-positive two for the result.  Finally, a
positive number and a contra-positive, with equal quotities,
(such as positive three and contra-positive three,) destroy each
other by addition; that is, they generate as their algebraic sum
the number None or Zero.

It is unnecessary to dwell on the algebraical operation of
{\it decomposition of multiple steps}, and consequently of whole
or {\it multipling numbers}, which corresponds to and includes
the operation of arithmetical {\it subtraction\/}; since it
follows manifestly from the foregoing articles of this Essay,
that the decomposition of numbers (like that of steps) can always
be performed by {\it compounding\/} with the given compound
number (that is, by algebraically {\it adding\/} thereto) the
{\it opposite\/} or opposites of the given component or
components: the number or numbers proposed to be subtracted are
therefore either to be neglected if they be null, since in that
case they have no effect, or else to be changed from positive to
contra-positive, or from contra-positive to positive, (their
quotities being preserved,) and then added (algebraically) in
this altered state.  Thus, positive five is subtracted
algebraically from positive two by adding contra-positive five,
and the result is contra-positive three; that is, the given step
$2 \times{\sr a}$ or $2 {\sr a}$ may be decomposed into two
others, of which the given component step $5 \times {\sr a}$ is
one, and the sought component step $3 \oppos {\sr a}$ is the
other.

\bigbreak

15.
Any multiple step~$\mu {\sr a}$ may be treated as a new base, or
new unit-step; and thus we may generate from it a new system of
multiple steps.  It is evident that these multiples of a multiple
of a step are themselves multiples of that step; that is, if we
first multiple a given base or unit-step~${\sr a}$ by any whole
number~$\mu$, and then again multiple the result
$\mu \times {\sr a}$ by any other whole number~$\nu$, the final
result $\nu \times (\mu \times {\sr a})$ will necessarily be of
the form $\omega \times {\sr a}$, $\omega$ being another whole
number.  It is easy also to see that the new multipling number,
such as $\omega$, of the new or derived multiple, must depend on
the old or given multipling numbers, such as $\mu$ or $\nu$, and
on these alone; and the law of its dependence on them may be
conveniently expressed by the same mark of combination~$\times$
which we have already used to combine any multipling number with
its base; so that we may agree to write
$$\omega = \nu \times \mu,
   \quad\hbox{when}\quad
      \omega \times {\sr a} = \nu \times (\mu \times {\sr a}).
   \eqno (111.)$$
With this definition of the effect of the combining
sign~$\times$, when interposed between the signs of two whole
numbers, we may write
$$\nu \times (\mu \times {\sr a})
   = (\nu \times \mu) \times {\sr a}
   = \nu \times \mu \times {\sr a},
   \eqno (112.)$$
omitting the parentheses as unnecessary; because, although their
absence permits us to interpret the complex symbol
$\nu \times \mu \times {\sr a}$ either as
$\nu \times (\mu \times {\sr a})$ or as
$(\nu \times \mu) \times {\sr a}$,
yet both the processes of combination thus denoted conduct to one
common result, or ultimate step.  (Compare article~11.)

When $\mu$ and $\nu$ are positive numbers, the law of combination
expressed by the notation $\nu \times \mu$, as above explained,
is easily seen to be that which is called {\it Multiplication\/}
in elementary Arithmetic, namely, the arithmetical addition of a
given number~$\nu$ of equal quotities~$\mu$; and the resulting
quotity $\nu \times \mu$ is the arithmetical {\it product\/} of
the numbers to be combined, or the product of $\mu$ multiplied by
$\nu$: thus we must, by the definition (112.), interpret
$3 \times 2$ as denoting the positive number~$6$, because
$3 \times (2 \times {\sr a})= 6 \times {\sr a}$, the triple of
the double of any step~${\sr a}$ being the sextuple of that step;
and the quotity~$6$ is, for the same reason, the arithmetical
product of $2$ multiplied by $3$, in the sense of being the
answer to the question, How many things or thoughts (in this
case, steps) are contained in a total group, if that total group
be composed of $3$ partial groups, and if $2$ such things or
thoughts be contained in each of these?  From this analogy to
arithmetic, we may in general call $\nu \times \mu$ the
{\it product\/} or (more fully) the {\it algebraic product}, of
the whole numbers $\mu$ and $\nu$, whether these, which we may
call the {\it factors\/} of the product, be positive, or
contra-positive, or null; and may speak of the process of
combination of these numbers, as the {\it multipling}, or (more
fully) the {\it algebraic multipling\/} of $\mu$ by $\nu$:
reserving still the more familiar arithmetical word
``multiplying'' to be used in algebra in a more general sense,
which includes the operation of multipling, and which there will
soon be occasion to explain.

In like manner, three or more whole numbers, $\mu$, $\nu$, $\xi$,
may be used successively to multiple a given step or one another,
and so to generate a new derived multiple of the original step or
number; thus, we may write
$$\xi \times \{ \nu \times (\mu \times {\sr a}) \}
   = \xi \times \{ (\nu \times \mu) \times {\sr a} \}
   = (\xi \times \nu \times \mu) \times {\sr a},
   \eqno (113.)$$
the symbol $\xi \times \nu \times \mu$ denoting here a new whole
number, which may be called the algebraic {\it product\/} of the
{\it three\/} whole numbers $\mu$, $\nu$, $\xi$, those numbers
themselves being called the {\it factors\/} of this product.
With respect to the actual process of such {\it multipling}, or
the rules for forming such algebraic {\it products\/} of whole
numbers, (whether positive, or contra-positive, or null,) it is
sufficient to observe that the product is evidently null if any
one of the factors be null, but that otherwise the product is
contra-positive or positive, according as there is or is not an
odd number (such as one, or three, or five, \&c.) of
contra-positive factors, because the direction of a step is not
changed, or is restored, when it is either not reversed at all,
or reversed an even number of times; and that, in every case, the
quotity of the algebraic product is the arithmetical product of
the quotities of the factors.  Hence, by the properties of
arithmetical products, or by the principles of the present essay,
we see that in forming an algebraical product the order of the
factors may be altered in any manner without altering the result,
so that
$$\nu \times \mu = \mu \times \nu,\quad
  \xi \times \nu \times \mu = \mu \times \xi \times \nu
  = \hbox{\&c., \&c.};
   \eqno (114.)$$
and that any one of the factors may be decomposed in any manner
into algebraical parts or component whole numbers, according to
the rules of algebraic addition and subtraction of whole numbers,
and each part separately combined as a factor with the other
factors to form a partial product, and then these partial
products algebraically added together, and that the result will
be the total product; that is,
$$\left. \eqalign{
(\nu' + \nu) \times \mu
   &= (\nu' \times \mu) + (\nu \times \mu),\cr
\nu \times (\mu' + \mu)
   &= (\nu \times \mu') + (\nu \times \mu),\quad\hbox{\&c.}\cr}
   \right\}
   \eqno (115.)$$

Again, we saw that if a factor~$\mu$ be null, the product is then
null also,
$$\nu \times 0 = 0;
   \eqno (116.)$$
because the multiples of a null multiple step are all themselves
null steps.  But if, in a product of two whole numbers,
$\nu \times \mu$, the first factor~$\mu$ (with which by (114.)
the second factor~$\nu$ may be interchanged) be given, and
effective, that is, if it be any given positive or
contra-positive whole number, ($\mu \neq 0$,) then its several
multiples, or the products of the form $\nu \times \mu$, form an
indefinite series of whole numbers,
$$\ldots \,
   3 \oppos \times \mu,\quad
   2 \oppos \times \mu,\quad
   1 \oppos \times \mu,\quad
   0 \times \mu,\quad
   1 \times \mu,\quad
   2 \times \mu,\quad
   3 \times \mu,\ldots
   \eqno (117.)$$
such that any proposed whole number~$\omega$, whatever, must be
either a multiple of this series, or else included between two
successive numbers of it, such as $\nu \times \mu$ and
$(1 + \nu) \times \mu$, being on the positive side of one of
them, and on the contra-positive side of the other, in the
complete series of whole numbers (103.).  In the one case, we can
satisfy the equation
$$\omega = \nu \times \mu,
   \quad\hbox{or,}\quad
   \oppos (\nu \times \mu) + \omega = 0,
   \eqno (118.)$$
by a suitable choice of the whole number~$\nu$; in the other
case, we cannot indeed do this, but we can choose a whole
number~$\nu$, such that
$$\omega = \rho + (\nu + \mu),
   \quad\hbox{or,}\quad
  \oppos (\nu \times \mu) + \omega = \rho,
   \eqno (119.)$$
$\rho$ being a whole number which lies between $0$ and $\mu$ in
the general series of whole numbers (103.), and which therefore
has a quotity less than the quotity of that given first
factor~$\mu$, and is positive or contra-positive according as
$\mu$ is positive or contra-positive.  In each case, we may be
said (by analogy to arithmetical division) to have
{\it algebraically divided\/} (or rather {\it measured\/}),
accurately or approximately, the whole number~$\omega$ by the
whole number~$\mu$, and to have found a whole number~$\nu$ which
is either the {\it accurate quotient\/} (or {\it measure\/}) as
in the case (118.), or else the {\it next preceding integer}, as
in the other case (119.); in which last case the whole
number~$\rho$ may be called the {\it remainder\/} of the division
(or of the {\it measuring\/}).  In this second case, namely, when
it is impossible to perform the division, or the {\it measuring},
exactly, in whole numbers, because the proposed {\it dividend},
or {\it mensurand}, $\omega$, is not contained among the series
(117.) of multiples of a proposed {\it divisor}, or
{\it measurer},~$\mu$, we may choose to consider as the
approximate integer {\it quotient}, or {\it measure}, the
{\it next succeeding\/} whole number $1+ \nu$, instead of the
next preceding whole number~$\nu$; and then we shall have a
different {\it remainder}, $\oppos \mu + \rho$, such that
$$\omega = (\oppos \mu + \rho) + (\overline{1 + \nu} \times \mu),
   \eqno (120.)$$
which new remainder $\oppos \mu + \rho$ has still a quotity less
than that of $\mu$, but lies between $0$ and $\oppos \mu$,
instead of lying (like $\rho$) between $0$ and $\mu$, in the
general series of whole numbers (103.), and is therefore
contra-positive if $\mu$ be positive, or positive if $\mu$ be
contra-positive.  With respect to the actual process of
calculation, for discovering whether a proposed algebraical
division (or measuring), of one whole number by another, conducts
to an accurate integer quotient, or only to two approximate
integer quotients, a next preceding and a next succeeding, with
positive and contra-positive remainders; and for actually finding
the names of these several quotients and remainders, or their
several special places in the general series of whole numbers:
this algebraical process differs only by some slight and obvious
modifications (on which it is unnecessary here to dwell,) from
the elementary arithmetical operation of dividing one quotity by
another; that is, the operation of determining what multiple the
one is of the other, or between what two successive multiples it
is contained.  Thus, having decomposed by arithmetical division
the quotity~$8$ into the arithmetical sum of $1 \times 5$ and
$3$, and having found that it falls short by $2$ of the
arithmetical product $2 \times 5$, we may easily infer from hence
that the algebraic whole number {\it contra-positive eight\/} can
be only approximately measured (in whole numbers), as a
mensurand, by the measurer {\it positive five\/}; the next
succeeding integer quotient or measure being {\it contra-positive
one}, with {\it contra-positive three\/} for remainder, and the
next preceding integer quotient or measure being
{\it contra-positive two}, with {\it positive two\/} as the
remainder.  It is easy also to see that this algebraic measuring
of one whole number by another, corresponds to the accurate or
approximate measuring of one step by another.  And in like manner
may all other arithmetical operations and reasonings upon
quotities be generalised in Algebra, by the consideration of
multiple steps, and of their connected positive and
contra-positive and null whole numbers.

\bigbreak

\centerline{\it
On the Sub-multiples and Fractions of any given Step in the
Progression of Time; on the}

\nobreak\vskip 3pt

\centerline{\it
Algebraic Addition, Subtraction, Multiplication, and Division, of
Reciprocal and Fractional}

\nobreak\vskip 3pt

\centerline{\it
Numbers, positive and contra-positive; and on the impossible or
indeterminate act of}

\nobreak\vskip 3pt

\centerline{\it
sub-multipling or dividing by zero.}

\nobreak\bigskip

16.
We have seen that from the thought of any one step~${\sr a}$, as
a base or unit-step, we can pass to the thought of a series or
system of multiples of that base, namely, the series (98.) or
(99.) or (104.), having each a certain relation of its own to the
base, as such or such a particular multiple thereof, or as
mentally generated from that base by such or such a particular
act of multipling; and that every such particular relation, and
every such particular act of multipling, may be distinguished
from all such other relations, and from all such other acts, in
the entire series or system of these relations, and in the entire
system of these acts of multipling, by its own special or
determining whole number, whether ordinal or cardinal, and
whether positive, or contra-positive, or null.  Now every such
relation or act must be conceived to have a certain inverse or
reciprocal, by which we may, in thought, connect the base with
the multiple, and return to the former from the latter: and,
generally, the conception of passing (in thought) from a base or
unit-step to any one of its multiples, or of returning from the
multiple to the base, is included in the more comprehensive
conception of passing from any one such multiple to any other;
that is, from any one step to any other step
{\it commensurable\/} therewith, two steps being said to be
{\it commensurable\/} with each other when they are multiples of
one common base or unit-step, because they have then that common
base or unit for the {\it common measurer}.  The base, when thus
compared with one of its own multiples, may be called a
{\it sub-multiple\/} thereof; and, more particularly, we may call
it the ``second positive sub-multiple'' of its own second
positive multiple, the ``first contra-positive sub-multiple'' of
its own first contra-positive multiple, and so forth; retaining
always, to distinguish any one sub-multiple, the determining
ordinal of the multiple to which it corresponds: and the act of
returning from a multiple to the base, may be called an act of
{\it sub-multipling\/} or (more fully) of sub-multipling
{\it by\/} the same determining cardinal number by which the base
had been multipled before; for example, we may return to the base
from its second contra-positive multiple, by an act of thought
which may be called sub-multipling by contra-positive two.  Some
particular sub-multiples, and acts of sub-multipling, have
particular and familiar names; thus, the second positive
sub-multiple of any given step, and the act of sub-multipling a
given step by positive two, may be more familiarly described as
the {\it half\/} of that given step, and as the act of
{\it halving\/} it.  And the more comprehensive conception above
mentioned, of the act of passing from any one step~${\sr b}$ to
any other step~${\sr c}$ commensurable therewith, or from any one
to any other multiple of one common measure, or base, or
unit-step~${\sr a}$, may evidently be resolved into the foregoing
conceptions of the acts of multipling and sub-multipling; since
we can always pass first by an act of sub-multipling from the
given step~${\sr b}$, considered as a multiple of the
base~${\sr a}$, to that base~${\sr a}$ itself, as an auxiliary or
intermediate thought, and then proceed, by an act of multipling,
from this auxiliary thought or step, to its other
multiple~${\sr c}$.  Any one step~${\sr c}$ may therefore be
considered as a multiple of a sub-multiple of any other
step~${\sr b}$, if those two steps be commensurable; and the act
of passing from the one to the other is an act compounded of
sub-multipling and multipling.

Now, all acts thus compounded, besides the acts of multipling and
sub-multipling themselves, (and other acts, to be considered
afterwards, which may be regarded as of the same kind with these,
being connected with them by certain intimate relations, and by
one common character,) may be classed in algebra under the
general name of {\it multipling acts}, or acts of {\it algebraic
multiplication\/}; the {\it object\/} on which any such
{\it act\/} operates being called the {\it multiplicand}, and the
{\it result\/} being called the {\it product\/}; while the
{\it distinct thought\/} or {\it sign\/} of such an act is called
the {\it algebraic multiplier}, or {\it multiplying number\/}:
whatever this distinctive thought or sign may be, that is,
whatever conceived, or spoken, or written {\it specific rule\/}
it may involve, for specifying one particular act of
multiplication, and for distinguishing it from every other.  The
relation of an algebraic product to its algebraic multiplicand
may be called, in general, {\it ratio}, or {\it algebraic
ratio\/}; but the particular ratio of any one particular product
to its own particular multiplicand, depends on the particular act
of multiplication by which the one may be generated from the
other: the {\it number\/} which specifies the {\it act\/} of
multiplication, serves therefore also to specify the resulting
{\it ratio\/}, and every number may be viewed either as the
{\it mark of a ratio}, or as a {\it mark of a multiplication},
according as we conceive ourselves to be {\it analytically
examining\/} a product already formed, or {\it synthetically
generating\/} that product.

We have already considered that series or system of
{\it algebraic integers}, or {\it whole numbers}, (positive,
contra-positive, or null,) which mark the several possible ratios
of all multiple steps to their base, and the several acts of
multiplication by which the former may be generated from the
latter; namely, all those several acts which we have included
under the common head of {\it multipling}.  The inverse or
reciprocal acts of {\it sub-multipling}, which we must now
consider, and which we have agreed to regard as comprehended
under the more general head of {\it multiplication}, conduct to a
new class of multiplying numbers, which we may call
{\it reciprocals of whole numbers}, or, more concisely,
{\it reciprocal numbers\/}; and to a corresponding class of
ratios, which we may call {\it reciprocals of integer ratios}.
And the more comprehensive conception of the act of passing from
one to another of any two commensurable steps, conducts to a
correspondingly extensive class of multiplying acts, and
therefore also of multiplying numbers, and of ratios, which we
may call {\it acts of fractioning}, and {\it fractional numbers},
or {\it fractional ratios\/}; while the {\it product\/} of any
such act of fractioning, or of multiplying by any such fractional
number, that is, the {\it generated step\/} which is any multiple
of any sub-multiple of any proposed step or {\it multiplicand},
may be called a {\it fraction\/} of that step, or of that
multiplicand.  A fractional number may therefore always be
determined, in thought and in expression, by {\it two whole
numbers}, namely the sub-multipling number, called also the
{\it denominator}, and the multipling number, called also the
{\it numerator}, (of the fraction or fractional number,) which
mark the two successive or component acts that make up the
complex act of fractioning.  Hence also the reciprocal number, or
reciprocal of any proposed whole number, which marks the act of
multiplication conceived to be equivalent to the act of
sub-multipling by that whole number, coincides with the
fractional number which has the same whole number for its
denominator, and the number~$1$ for its numerator, because a step
is not altered when it is multipled by positive one.  And any
whole number itself, considered as the mark of any special act of
multipling, may be changed to a fractional number with positive
one for its denominator, and with the proposed whole number for
its numerator; since such a fractional number, considered as the
mark of a special act of multiplication, is only the complex mark
of a complex act of thought equivalent to the simpler act of
multipling by the numerator of the fraction; because the other
component act, of sub-multipling by positive one, produces no
real alteration.  Thus, the conceptions of whole numbers, and of
reciprocal numbers, are included in the more comprehensive
conception of fractional numbers; and a complete theory of the
latter would contain all the properties of the former.

\bigbreak

17.
To form now a notation of fractions, we may agree to denote a
fractional number by writing the numerator over the denominator,
with a bar between; that is, we may write
$${\sr c} = {\nu \over \mu} {\sr b},
   \quad\hbox{or more fully,}\quad
  {\sr c} = {\nu \over \mu} \times {\sr b},
   \eqno (121.)$$
when we wish to express that two commensurable steps, ${\sr b}$
and ${\sr c}$, (which we shall, for the present, suppose to be
both effective steps,) may be severally formed from one common
base or unit-step~${\sr a}$, by multipling that base by the two
(positive or contra-positive) whole numbers $\mu$ and $\nu$, so
that
$${\sr b} = \mu \times {\sr a},\quad
  {\sr c} = \nu \times {\sr a}.
   \eqno (122.)$$

[We shall suppose throughout the whole of this and of the two
next following articles, that all the steps are effective, and
that all the numerators and denominators are positive or
contra-positive, excluding for the present the consideration of
null steps, and of null numerators or null denominators.]

Under these conditions, the step~${\sr c}$ is a fraction of
${\sr b}$, and bears to that step~${\sr b}$ the fractional ratio
$\displaystyle {\nu \over \mu}$, called also ``the ratio of $\nu$
to $\mu$;'' and ${\sr c}$ may be deduced or generated as a
product from ${\sr b}$ by a corresponding act of fractioning,
namely, by the act of multiplying ${\sr b}$ as a multiplicand by
the fractional number $\displaystyle {\nu \over \mu}$ as a
multiplier, or finally by the complex act of first submultipling
${\sr b}$ by the denominator~$\mu$, and then multipling the
result ${\sr a}$ by the numerator~$\nu$.  Under the same
conditions, it is evident that we may return from ${\sr c}$ to
${\sr b}$ by an inverse or reciprocal act of fractioning, namely,
by that new complex act which is composed of submultipling
instead of multipling by $\nu$, and then multipling instead of
submultipling by $\mu$; so that
$${\sr b} = {\mu \over \nu} \times {\sr c},
   \quad\hbox{when}\quad
      {\sr c} = {\nu \over \mu} \times {\sr b}:
   \eqno (123.)$$
on which account we may write
$${\sr b} = {\mu \over \nu} \times
         \left( {\nu \over \mu} \times {\sr b} \right),
   \quad\hbox{and}\quad
  {\sr c} = {\nu \over \mu} \times
         \left( {\mu \over \nu} \times {\sr c} \right),
   \eqno (124.)$$
whatever (effective) steps may be denoted by ${\sr b}$ and
${\sr c}$, and whatever (positive or contra-positive) whole
numbers may be denoted by $\mu$ and $\nu$.  The two acts of
fractioning, marked by the two fractional numbers
$\displaystyle {\nu \over \mu}$ and
$\displaystyle {\mu \over \nu}$,
are therefore opposite or {\it reciprocal acts}, of which each
destroys or undoes the effect of the other; and the fractional
numbers themselves may be called {\it reciprocal fractional
numbers}, or, for shortness, {\it reciprocal fractions\/}: to
mark which reciprocity we may use a new symbol~$\recip$, (namely,
the initial letter of the word Reciprocatio, distinguished from
the other uses of this same letter by being written in an
inverted position,) that is, we may write
$${\nu \over \mu} = \recip {\mu \over \nu},\quad
  {\mu \over \nu} = \recip {\nu \over \mu},
   \eqno (125.)$$
whatever positive or contra-positive whole numbers may be marked
by $\mu$ and $\nu$.  In this notation,
$$\recip \recip {\nu \over \mu}
   = \recip \left( \recip {\nu \over \mu} \right)
   = \recip {\mu \over \nu}
   = {\nu \over \mu};
   \eqno (126.)$$
or, to express the same thing in words, the reciprocal of the
reciprocal of any fractional number is that fractional number
itself.  (Compare equation (57.)).

It is evident also, that
$${\sr a} = {1 \over \mu} \times {\sr b},
   \quad\hbox{and}\quad
  {\sr b} = {\mu \over 1} \times {\sr a},
   \quad\hbox{if}\quad
  {\sr b} = \mu \times {\sr a};
   \eqno (127.)$$
that is, the whole number~$\mu$, regarded as a multiplier, or as
a ratio, may be put under the fractional form
$\displaystyle {\mu \over 1}$, so that we may write
$${\mu \over 1} = \mu;
   \eqno (128.)$$
and the reciprocal of this whole number, or the connected
reciprocal number $\recip \mu$ to multiply by which is equivalent
to submultipling by $\mu$, coincides with the reciprocal
fractional number
$\displaystyle {1 \over \mu}$,
so that
$${1 \over \mu} = \recip {\mu \over 1} = \recip \mu:
   \eqno (129.)$$
remarks which were indeed anticipated in the remarks made at the
close of the foregoing article, respecting the extent of the
conception of fractional numbers, as including whole numbers and
their reciprocals.  As an example of these results, the double of
any step~${\sr a}$ may be denoted by the symbol
$\displaystyle {2 \over 1} \times {\sr a}$
as well as by $2 \times {\sr a}$, and the half of that
step~${\sr a}$ may be denoted either by the symbol
$\displaystyle {1 \over 2} \times {\sr a}$, or by
$\recip 2 \times {\sr a}$.  The symbol~$\recip 1$ is evidently
equivalent to $1$, the number positive one being its own
reciprocal; and the opposite number, contra-positive one, has the
same property, because to reverse the direction of a step is an
act which destroys itself by repetition, leaving the last
resulting step the same as the original; we have therefore the
equations,
$$\recip 1 = 1,\quad \recip \oppos 1 = \oppos 1.
   \eqno (130.)$$

By the definition of a fraction, as a multiple of a submultiple,
we may now express it as follows:
$${\nu \over \mu} \times {\sr b}
   = \nu \times \left( {1 \over \mu} \times {\sr b} \right)
   = \nu \times ( \recip \mu \times {\sr b} ).
   \eqno (131.)$$
Besides, under the conditions (122.), we have, by (122.) and
(114.), that is, by the principle of the indifference of the
order in which any two successive multiplings are performed,
$$\mu \times {\sr c}
   =  \mu \times ( \nu \times {\sr a} )
   =  (\mu \times \nu ) \times {\sr a}
   =  (\nu \times \mu ) \times {\sr a}
   =  \nu \times ( \mu \times {\sr a} )
   =  \nu \times {\sr b};
   \eqno (132.)$$
so that a fractional product
$\displaystyle {\sr c} = {\nu \over \mu} \times {\sr b}$
may be derived from the multiplicand~${\sr b}$, by first
multipling by the numerator~$\nu$ and then submultipling by the
denominator~$\mu$, instead of first submultipling by the latter
and afterwards multipling by the former; that is, in any act of
fractioning, we may change the order of the two successive and
component acts of submultipling and multipling, without altering
the final result, and may write
$${\nu \over \mu} \times {\sr b}
   =  {1 \over \mu} \times (\nu \times {\sr b})
   =  \recip \mu \times (\nu \times {\sr b}).
   \eqno (133.)$$
In general it may easily be shown, by pursuing a reasoning of the
same sort, that in any set of acts of multipling and
submultipling, to be performed successively on any one original
step, the order of succession of those acts may be altered in any
arbitrary manner, without altering the final result.  We may
therefore compound any proposed set of successive acts of
fractioning, by compounding first the several acts of
submultipling by the several denominators into the one act of
submultipling by the product of those denominators, and then the
several acts of multipling by the several numerators into the one
act of multipling by the product of those numerators, and finally
the two acts thus derived into one last resultant act of
fractioning; that is, we have the relations,
$$\left. \eqalign{
{\nu' \over \mu'} \times
      \left( {\nu \over \mu} \times {\sr b} \right)
   &= {\nu' \times \nu \over \mu' \times \mu} \times {\sr b},\cr
{\nu'' \over \mu''} \times
      \left\{ {\nu' \over \mu'} \times
      \left( {\nu \over \mu} \times {\sr b} \right) \right\}
   &= {\nu'' \times \nu' \times \nu \over \mu'' \times \mu' \times \mu}
         \times {\sr b},\cr
\omit\hfil\hbox{\&c.}\hfil\cr}
   \right\}
   \eqno (134.)$$
We may also introduce or remove any positive or contra-positive
whole number as a factor in both the numerator and the
denominator of any fraction, without making any real alteration;
that is, the following relation holds good:
$${\nu \over \mu} = {\omega \times \nu \over \omega \times \mu},
   \eqno (135.)$$
whatever positive or contra-positive whole numbers may be denoted
by $\mu~\nu~\omega$; a theorem which may often enable us to put a
proposed fraction under a form more simple in itself, or more
convenient for comparison with others.  As particular cases of
this theorem, corresponding to the case when the common
factor~$\omega$ is contra-positive one, we have
$${\nu \over \mu} = {\oppos \nu \over \oppos \mu},\quad
  {\oppos \nu \over \mu} = {\nu \over \oppos \mu};
   \eqno (136.)$$
that is, the denominator of any fraction may be changed from
contra-positive to positive, or from positive to contra-positive,
without making any real change, provided that the numerator is
also changed to its own opposite whole number.  Two fractional
numbers, such as
$\displaystyle {\oppos \nu \over \mu}$ and
$\displaystyle {\nu \over \mu}$,
may be said to be {\it opposites}, (though {\it not
reciprocals\/}), when (through {\it not\/} themselves the marks
of {\it opposite acts\/}), they {\it generate opposite steps},
such as the steps
$\displaystyle {\oppos \nu \over \mu} \times {\sr b}$ and
$\displaystyle {\nu \over \mu} \times {\sr b}$;
and to mark this opposition we may write
$${\oppos \nu \over \mu} = \oppos {\nu \over \mu}.
   \eqno (137.)$$
Hence every fractional number, with any positive or
contra-positive whole numbers $\mu$ and $\nu$ for its denominator
and numerator, may be put under one or other of the two following
forms:
$$\hbox{Ist.}\quad
      {n \over m},
   \quad\hbox{or}\quad\hbox{IInd.}\quad
      \oppos {n \over m},
   \eqno (138.)$$
($m$ and $n$ denoting positive whole numbers,) according as the
proposed whole numbers $\mu$ and $\nu$ agree or differ in respect
of being positive or contra-positive; and in the Ist case we may
say that the fractional number itself is {\it positive}, but in
the IInd case that it is {\it contra-positive\/}: definitions
which agree with and include the former conceptions of positive
and contra-positive whole numbers, when we consider these as
equivalent to fractional numbers in which the numerator is a
multiple of the denominator; and lead us to regard the reciprocal
of any positive or contra-positive whole number (and more
generally the reciprocal of any positive or contra-positive
fractional number) as positive or contra-positive like it; a
fractional number being equivalent to the reciprocal of a whole
number, when the denominator is a multiple of the numerator.  A
fraction of a late-making step~${\sr b}$ is itself a late-making
or an early-making step, according as the multiplying fractional
number is positive or contra-positive; and as we have agreed to
write ${\sr b} > 0$ when ${\sr b}$ is a late-making step, so we
may now agree to write
$${\nu \over \mu} > 0,
   \quad\hbox{when}\quad
  {\nu \over \mu} \times {\sr b} > 0
   \quad\hbox{and}\quad
  {\sr b} > 0,
   \eqno (139.)$$
that is, when $\displaystyle {\nu \over \mu}$ is a {\it positive
fractional number}, and to write, on the contrary,
$${\nu \over \mu} < 0,
   \quad\hbox{when}\quad
  {\nu \over \mu} \times {\sr b} < 0
   \quad\hbox{and}\quad
  {\sr b} > 0,
   \eqno (140.)$$
that is, when
$\displaystyle {\nu \over \mu}$ is a {\it contra-positive
fractional number}.  More generally, we shall write
$${\nu' \over \mu'} > {\nu \over \mu},
   \quad\hbox{if}\quad
  {\nu' \over \mu'} \times {\sr b} > {\nu \over \mu} \times {\sr b},
   \quad {\sr b} > 0,
   \eqno (141.)$$
and
$${\nu' \over \mu'} < {\nu \over \mu},
   \quad\hbox{if}\quad
  {\nu' \over \mu'} \times {\sr b} < {\nu \over \mu} \times {\sr b},
   \quad {\sr b} > 0;
   \eqno (142.)$$
and shall enunciate these two cases respectively, by saying that
in the first case the fractional number
$\displaystyle {\nu' \over \mu'}$
is {\it on the positive side}, and that in the second case it is
{\it on the contra-positive side}, of the other fractional number
$\displaystyle {\nu \over \mu}$;
or that in the first case
$\displaystyle {\nu' \over \mu'}$
{\it follows\/} and that in the second it {\it precedes\/}
$\displaystyle {\nu \over \mu}$,
in the general progression of numbers, from contra-positive to
positive; definitions which may easily be shown to be consistent
with each other, and which extend to whole numbers and their
reciprocals, as included in fractional numbers, and to the number
zero itself as compared with any of these.  Thus, every positive
number is on the positive side of zero and of every
contra-positive number; while zero is on the positive side of all
contra-positive numbers, but on the contra-positive side of all
positive numbers: for example,
$$2 > 0,\quad 2 > \oppos 3,\quad \oppos 3 < 0,\quad
  \oppos 3 < 2,\quad 0 > \oppos 3,\quad 0 < 2.
   \eqno (143.)$$
Of two unequal positive whole numbers, the one which has the
greater quotity is on the positive side, but among
contra-positive numbers the reverse is the case; for example,
$$3 > 2,\quad \oppos 3 < \oppos 2:
   \eqno (144.)$$
and in general a relation of subsequence or precedence between
any two whole or fractional numbers is changed to the opposite
relation of precedence or subsequence, by altering those numbers
to their opposites, though a relation of equality or coincidence
remains unaltered after such a change.  Among reciprocals of
positive whole numbers, the reciprocal of that which has the
lesser quotity is on the positive side of the other, while
reciprocals of contra-positive numbers are related by the
opposite rule; thus
$${1 \over 2} > {1 \over 3},\quad
  {1 \over \oppos 2} < {1 \over \oppos 3},
   \quad\hbox{that is,}\quad
  \recip 2 > \recip 3,\quad
  \recip \oppos 2 < \recip \oppos 3.
   \eqno (145.)$$
In general, to determine the ordinal relation of any one
fractional number
$\displaystyle {\nu' \over \mu'}$
to another
$\displaystyle {\nu \over \mu}$,
as subsequent, or coincident, or precedent, in the general
progression of numbers, it is sufficient to prepare them by the
principle (135.) so that their denominators may be equal and
positive, and then to compare their numerators; for which reason
it is always sufficient to compare the two whole numbers
$\mu \times \mu \times \mu' \times \nu'$ and
$\mu' \times \mu' \times \mu \times \nu$,
and we have
$${\nu' \over \mu'} \gteqlt {\nu \over \mu},
   \quad\hbox{according as}\quad
   \mu \times \mu \times \mu' \times \nu'
      \gteqlt \mu' \times \mu' \times \mu \times \nu:
   \eqno (146.)$$
the abridged notation $\gteqlt$ implying the same thing as if
we had written more fully ``$>$ or $=$ or $<$.''  If it had been
merely required to prepare two fractional numbers so as to make
them have a common denominator, without obliging that denominator
to be positive, we might have done so in a simpler manner by the
formula (135.), namely by multiplying the numerator and
denominator of each fraction by the denominator of the other
fraction, that is, by employing the following expressions,
$${\nu' \over \mu'} = {\mu \times \nu' \over \mu \times \mu'},\quad
  {\nu \over \mu}   = {\nu \times \mu' \over \mu \times \mu'};
   \eqno (147.)$$
a process which may be still farther simplified when the original
denominators have any whole number (other than positive or
contra-positive one) for a common factor, since it is sufficeint
then to multiple by the factors which are not thus common,
that is, to employ the expressions,
$${\nu' \over \omega \times \mu'}
   = {\mu \times \nu' \over \omega \times \mu \times \mu'},\quad
  {\nu \over \omega \times \mu}
   = {\nu \times \mu' \over \omega \times \mu \times \mu'}.
   \eqno (148.)$$
A similar process of preparation applies to more fractions than
two.

\bigbreak

18.
This reduction of different fractional numbers to a common
denominator is chiefly useful in combining them by certain
operations which may be called {\it algebraical addition and
subtraction of fractions}, (from their analogy to the algebraical
addition and subtraction of whole numbers, considered in the 14th
article, and to the arithmetical operations of addition and
subtraction of quotities,) and which present themselves in
considering the composition and decomposition of fractional
steps.  For if we compound, as successive steps, any two or more
fractions
$\displaystyle {\nu \over \mu} \times {\sr b}$,
$\displaystyle {\nu' \over \mu'} \times {\sr b}$,
\&c., of any one effective step~${\sr b}$, and generate thereby a
new effective step, this resultant step will evidently be itself
a fraction of the step~${\sr b}$, which we may agree to denote as
follows:
$$\left. \eqalign{
         \left( {\nu'  \over \mu'}  \times {\sr b} \right)
       + \left( {\nu   \over \mu}   \times {\sr b} \right)
   &=    \left(
            {\nu'  \over \mu'}
          + {\nu   \over \mu}
         \right)
            \times {\sr b},\cr
         \left( {\nu'' \over \mu''} \times {\sr b} \right)
       + \left( {\nu'  \over \mu'}  \times {\sr b} \right)
       + \left( {\nu   \over \mu}   \times {\sr b} \right)
   &=    \left(
            {\nu'' \over \mu''}
          + {\nu'  \over \mu'}
          + {\nu   \over \mu}
         \right)
            \times {\sr b},
      \quad\hbox{\&c.};\cr}
   \right\}
   \eqno (149.)$$
and the resultant fractional number
$\displaystyle {\nu' \over \mu'} + {\nu \over \mu}$ or
$\displaystyle {\nu'' \over \mu''} + {\nu' \over \mu'} + {\nu \over \mu}$
\&c.\ may be called the algebraical {\it sum\/} of the proposed
fractional numbers
$\displaystyle {\nu \over \mu}$,
$\displaystyle {\nu' \over \mu'}$,
$\displaystyle {\nu'' \over \mu''}$,
\&c.\ and may be said to be formed by algebraically
{\it adding\/} them together; definitions which agree with those
established in the 14th article, when the fractional numbers
reduce themselves to whole numbers.  If the denominators of the
proposed fractions be the same, it is sufficient to add the
numerators, because then the proposed fractional steps are all
multiples of one common sub-multiple of the common
unit-step~${\sr b}$, namely of that sub-multiple which is
determined by the common denominator; it is therefore sufficient,
in other cases, to prepare the fractions so as to satisfy this
condition of having a common denominator, and afterwards to add
the numerators so prepared, and to combine their sum as the new
or resulting numerator of the resulting fractional sum, with the
common denominator of the added fractions as the denominator of
the same fractional sum; which may, however, be sometimes
simplified by the omission of common factors, according to the
principle (135.).  Thus
$${\nu' \over \mu'} + {\nu \over \mu}
   =  {(\nu' \times \mu) + (\mu' \times \nu)
         \over \mu' \times \mu},
   \quad\hbox{or more concisely}\quad
  {\nu' \over \mu'} + {\nu \over \mu}
   =  {\nu' \mu + \mu' \nu \over \mu' \mu},
   \quad\hbox{\&c.};
   \eqno (150.)$$
for, as a general rule of algebraic notation, we may omit at
pleasure the mark of multiplication between any two simple
symbols of factors, (except the arithmetical signs $1$,~$2$,~$3$,
\&c.,) without causing any confusion; and when a product thus
denoted, by the mere juxta-position of its factors, (without the
mark~$\times$,) is to be combined with other symbols in the way
of addition, by the mark~$+$, it is not necessary to enclose that
symbol of a product in parentheses: although in this Elementary
Essay we have often used, and shall often use again, these
combining and enclosing marks, for greater clearness and
fullness.  It is evident that the addition of fractions may be
performed in any arbitrary order, because the order of
composition of the fractional steps is arbitrary.

The algebraical {\it subtraction\/} of one given fractional
number
$\displaystyle {\nu' \over \mu'}$
from another unequal fractional number
$\displaystyle {\nu \over \mu}$,
is an operation suggested by the decomposition of a given
compound fractional step
$\displaystyle {\nu \over \mu} \times {\sr b}$
into a given component fractional step
$\displaystyle {\nu' \over \mu'} \times {\sr b}$
and a sought component fractional step
$\displaystyle {\nu'' \over \mu''} \times {\sr b}$,
(these three steps being here supposed to be all effective:) and
it may be performed by compounding the opposite of the given
component step with the given compound step, or by algebraically
adding the opposite
$\displaystyle \oppos {\nu' \over \mu'}$
of the given fractional number
$\displaystyle {\nu' \over \mu'}$
to the other given fractional number
$\displaystyle {\nu \over \mu}$,
according to the rule (150.).  When we thus subtract one
fractional number from another with which it does not coincide,
the result is positive or contra-positive according as the
fraction from which we subtract is on the positive or
contra-positive side of the other; and thus we have another
general method, besides the rule (146.), for examining the
ordinal relation of any two unequal fractions, in the general
progression of numbers.  This ordinal relation between any two
fractional (or whole) numbers $\alpha$ and $\beta$, is not
altered by adding any fractional (or whole) number~$\gamma$ to
both, nor by subtracting it from both; so that
$$\gamma + \beta \gteqlt \gamma + \alpha,
   \quad\hbox{and}\quad
  \oppos \gamma + \beta \gteqlt \oppos \gamma + \alpha,
   \quad\hbox{according as}\quad
  \beta \gteqlt \alpha.
   \eqno (151.)$$

\bigbreak

19.
Again, the composition and decomposition of {\it successive acts
of fractioning\/} (instead of successive fractional
{\it steps\/}) conduct to algebraical operations of
{\it multiplication\/} and {\it division\/} of fractional
numbers, which are analogous to the arithmetical operations of
multiplication and division of quotities.  For if we first
multiply a given step~${\sr b}$ by a given fractional number
$\displaystyle {\nu \over \mu}$, that is, if we first perform on
${\sr b}$ the act of fractioning denoted by this number, and so
form the fractional step
$\displaystyle {\nu \over \mu} \times {\sr b}$,
we may then perform on the result another act of fractioning
denoted by another fractional number
$\displaystyle {\nu' \over \mu'}$,
and so deduce another fractional step
$\displaystyle
   {\nu' \over \mu'} \times
      \left( {\nu \over \mu} \times {\sr b} \right)$,
which will evidently be itself a fraction of the original
step~${\sr b}$, and might therefore have been deduced from
${\sr b}$ by one compound act of fractioning; and thus we may
proceed to other and other fractions of that step, and to other
compound acts of fractioning, which may be thus denoted,
$$\left. \eqalign{
{\nu' \over \mu'} \times
      \left ({\nu \over \mu} \times {\sr b} \right)
   &= \left(
         {\nu' \over \mu'} \times
         {\nu \over \mu}
      \right)
         \times {\sr b},\cr
{\nu'' \over \mu''} \times \left\{ {\nu' \over \mu'} \times
      \left ({\nu \over \mu} \times {\sr b} \right) \right\}
   &= \left(
         {\nu'' \over \mu''} \times
         {\nu' \over \mu'} \times
         {\nu \over \mu}
      \right)
         \times {\sr b},
         \quad\hbox{\&c.};\cr}
   \right\}
   \eqno (152.)$$
and the resultant fractional numbers
$\displaystyle {\nu' \over \mu'} \times {\nu \over \mu}$,
$\displaystyle {\nu'' \over \mu''} \times {\nu' \over \mu'}
      \times {\nu \over \mu}$,
\&c., which thus express the resultant acts of fractioning,
derived from the proposed component acts marked by the fractional
numbers
$\displaystyle {\nu \over \mu}$,
$\displaystyle {\nu' \over \mu'}$,
$\displaystyle {\nu'' \over \mu''}$, \&c.,
may be called the {\it algebraic products\/} of those proposed
fractional numbers, and may be said to be formed by
{\it algebraically multiplying\/} them as {\it fractional
factors\/} together; definitions which agree with the definitions
of product and multiplication already established for whole
numbers.  The same definitions shew that every fraction may be
regarded as the product of the numerator (as one factor) and the
reciprocal of the denominator (as another); and give, in general,
by (134.), the following rule for the calculation of a fractional
product
$${\nu' \over \mu'} \times {\nu \over \mu}
   =  {\nu' \times \nu \over \mu' \times \mu},\quad
  {\nu'' \over \mu''} \times {\nu' \over \mu'} \times {\nu \over \mu}
   =  {\nu'' \times \nu' \times \nu
         \over \mu'' \times \mu' \times \mu'},
   \quad\hbox{\&c.}
   \eqno (153.)$$
The properties (114.) and (115.) of algebraic products of whole
numbers extend to products of fractional numbers also; that is,
we may change in any manner the order of the fractional factors;
and if we resolve any one of those factors into two or more
algebraic parts by the rules of algebraic addition and
subtraction, we may combine each part separately as a partial
factor with the other factors proposed, so as to form by
algebraic multiplication a partial fractional product, and then
add together those partial products algebraically to obtain the
total product: or, in written symbols,
$${\nu' \over \mu'} \times {\nu \over \mu}
   =  {\nu \over \mu} \times {\nu' \over \mu'},
   \quad\hbox{\&c.},
   \eqno (154.)$$
and
$${\nu \over \mu} \times
         \left( {\nu'' \over \mu''} + {\nu' \over \mu'} \right)
   =     \left( {\nu \over \mu} \times {\nu'' \over \mu''} \right)
       + \left( {\nu \over \mu} \times {\nu'  \over \mu' } \right),
   \quad\hbox{\&c.},
   \eqno (155.)$$
because
$${\nu \over \mu} \times ({\sr b}'' + {\sr b}')
   =     \left( {\nu \over \mu} \times {\sr b}'' \right)
       + \left( {\nu \over \mu} \times {\sr b}'  \right),
   \eqno (156.)$$
whatever steps may be denoted by ${\sr b}'$ and ${\sr b}''$ and
whatever fractional (or whole) number by
$\displaystyle {\nu \over \mu}$.  We may also remark that
$$\gamma \times \beta \gteqlt \gamma \times \alpha,
   \quad\hbox{according as}\quad
  \beta \gteqlt \alpha,
   \quad\hbox{if}\quad \gamma > 0,
   \eqno (157.)$$
but that
$$\gamma \times \beta \lteqgt \gamma \times \alpha,
   \quad\hbox{according as}\quad
  \beta \gteqlt \alpha,
   \quad\hbox{if}\quad \gamma < 0,
   \eqno (158.)$$
$\alpha$~$\beta$~$\gamma$ denoting any three fractional (or
whole) numbers.

The deduction of one of two fractional factors from the other and
from the product, may be called (by analogy to arithmetic) the
{\it algebraic division\/} of the given fractional product as a
{\it dividend}, by the given fractional factor as a
{\it divisor\/}; and the result, which may be called the
{\it quotient}, may always be found by algebraically multiplying
the proposed dividend by the reciprocal of the proposed divisor.
This more general conception of quotient, agrees with the process
of the 15th article, for the division of one whole number by
another, when that process gives an accurate quotient in whole
numbers; and when no such integral and accurate quotient can be
found, we may still, by our present extended definitions,
conceive the numerator of any fraction to be divided by the
denominator, and the quotient of this division will be the
fractional number itself.  In this last case, the fractional
number is not exactly equal to any whole number, but lies between
two successive whole numbers, a next preceding and a next
succeeding, in the general progression of numbers; and these may
be discovered by the process of approximate division above
mentioned, while each of the two remainders of that approximate
division is the numerator of a new fraction, which retains the
proposed denominator, and must be added algebraically as a
{\it correction\/} to the corresponding {\it approximate integer
quotient}, in order to express, by the help of it, the quotient
of the accurate division.  For example,
$${8 \over 5}
   =  {3 \over 5} + 1
   =  {\oppos 2 \over 5} + 2,
   \quad\hbox{and}\quad
   {\oppos  8 \over 5}
   =  {2 \over 5} + \oppos 2
   =  {\oppos 3 \over 5} + \oppos 1.$$
In general, a fractional number may be called a {\it mixed
number}, when it is thus expressed as the algebraic sum of a
whole number and a {\it proper fraction}, this last name being
given to a fractional number which lies between zero and positive
or contra-positive one.  We may remark that an ordinal relation
beteen two fractional numbers is not altered by dividing them
both by one common positive divisor; but if the divisor be
contra-positive, it changes a relation of subsequence to one of
precedence, and conversely, without disturbing a relation of
coincidence.

\bigbreak

20.
In all the formul{\ae} of the last three articles, we have
supposed that all the numerators and all the denominators of
those formul{\ae} are positive or contra-positive whole numbers,
excluding the number zero.  However, the general conception of a
fraction as {\it a multiple of a sub-multiple}, permits us to
suppose that the multipling number or numerator is zero, and
shows us that then the fractional step itself is null, if the
denominator be different from zero; that is,
$${0 \over \mu} \times {\sr b} = 0
   \quad\hbox{if}\quad \mu \neq 0.
   \eqno (159.)$$
Thus, although we supposed, in the composition (149.) of
successive fractional steps, (with positive or contra-positive
numerators and denominators,) that the resultant step was
effective, yet we might have removed this limitation, and have
presented the formul{\ae} (150.) for fractional sums as extending
even to the case when the resultant step is null, if we had
observed that in every such case the resultant numerator of the
formula is zero, while the resultant denominator is different
from zero, and therefore that the formula rightly expresses that
the resultant fraction or sum is null.  For example, the addition
of any two opposite fractional numbers, such as
$\displaystyle {\nu \over \mu}$ and
$\displaystyle {\oppos \nu \over \mu}$,
in which $\mu$ and $\nu$ are different from zero, conducts to a
null sum, under the form
$\displaystyle {\oppos \nu + \nu \over \mu}$,
in which the numerator $\oppos \nu + \nu$ is zero, while the
denominator is different from zero.

But it is not so immediately clear what ought to be regarded as
the meaning of a fractional sign, in the case when the
denominator is null, and when therefore the act of fractioning
prescribed by the notation involves a sub-multipling by zero.  To
discuss this case, we must remember that to sub-multiple a
step~${\sr b}$ by a whole number~$\mu$, is, by its definition, to
find another step~${\sr a}$, which, when multipled by that whole
number~$\mu$, shall produce the proposed step~${\sr b}$; but,
whatever step~${\sr a}$ may be, the theory of multiple steps
(explained in the 13th article) shows that it necessarily
produces the null step~$0$, when it is multipled by the null
number zero; that is, the equation
$$0 \times {\sr a} = 0
   \eqno (160.)$$
is true independently of ${\sr a}$, and consequently we have
always
$$0 \times {\sr a} \neq {\sr b},
   \quad\hbox{if}\quad {\sr b} \neq 0.
   \eqno (161.)$$
It is, therefore, impossible to find any step~${\sr a}$, in the
whole progression of time, which shall satisfy the equation
$${1 \over 0} \times {\sr b} = {\sr a},
   \quad\hbox{or}\quad
  0 \times {\sr a} = {\sr b},
   \eqno (162.)$$
if the given step be effective; or, in other words, it is
impossible to sub-multiple an effective step by zero.  The
fractional sign
$\displaystyle {1 \over 0}$
denotes therefore an {\it impossible act}, if it be applied to an
effective step: and {\it the zero submultiple of an effective
step\/} is a phrase which involves a contradiction.  On the other
hand, if the given step~${\sr b}$ be null, it is not only
possible to choose some one step~${\sr a}$ which shall satisfy
the equations (162.), but every conceivable step possesses the
same proposed property: in this case, therefore, the proposed
conditions lay no restriction on the result, but at the same
time, and for the same reason, they fail to give any information
respecting it; and the act of sub-multipling a null step by zero,
is indeed a possible, but it is also an {\it indeterminate act},
or an act with an indeterminate result; so that the
{\it zero-submultiple of a null step}, and the written symbol
$\displaystyle {1 \over 0} \times 0$,
are spoken or written signs which do not specify any thing,
although they do not involve a contradiction.  We see then that
while a fractional number is in general the sign of a possible
and determinate act of fractioning, it loses one or other of
those two essential characters whenever its denominator is zero;
for which reason it becomes comparatively unfit, or at least
inconvenient, in this case, for the purposes of mathematical
reasoning.  And to prevent the confusion which might arise from
the mixture of such cases with others, it is convenient to lay
down this {\it general rule}, to which we shall henceforth
adhere: that {\it all denominators and divisors are to be
supposed different from zero\/} unless the contrary be mentioned
expressly; or that we shall {\it never sub-multiple nor divide by
a null-number\/} without expressly recording that we do so.

\bigbreak

\centerline{\it
On the Comparison of any one effective Step with any other, in
the way of Ratio, and the}

\nobreak\vskip 3pt

\centerline{\it
Generation of any one such step from any other, in the way of
Multiplication; and on the}

\nobreak\vskip 3pt

\centerline{\it
Addition, Subtraction, Multiplication, and Division of Algebraic
Numbers in general,}

\nobreak\vskip 3pt

\centerline{\it
considered thus as Ratios or as Multipliers of Steps.}

\nobreak\bigskip

21.
The foregoing remarks upon fractions lead naturally to the more
general conception of {\it algebraic ratio}, as a complex
relation of any one effective step to any other, determined by
their {\it relative largeness\/} and {\it relative direction\/};
and to a similarly extended conception of algebraic
{\it multiplication}, as an {\it act\/} (of thought) which
enlarges, or preserves, or diminishes the magnitude, while it
preserves or reverses the direction, of any effective step
proposed.  In conformity with these conceptions, and by analogy
to our former notations, if we denote by ${\sr a}$ and ${\sr b}$
any two effective steps, of which ${\sr a}$ may be called the
{\it antecedent\/} or the {\it multiplicand}, and ${\sr b}$ the
{\it consequent\/} or the {\it product}, we may employ the symbol
$\displaystyle {{\sr b} \over {\sr a}}$
to denote the {\it ratio\/} of the consequent~${\sr b}$ to the
antecedent~${\sr a}$, or the algebraic {\it number\/} or
{\it multiplier\/} by which we are to multiply ${\sr a}$ as a
{\it multiplicand\/} in order to generate ${\sr b}$ as a product:
and if we still employ the mark of multiplication~$\times$, we
may now write, in general,
$${\sr b} = {{\sr b} \over {\sr a}} \times {\sr a}:
   \eqno (163.)$$
or, more concisely,
$${\sr b} = a \times {\sr a},
   \quad\hbox{if}\quad
  {{\sr b} \over {\sr a}} = a,
   \eqno (164.)$$
that is, if we employ, for abridgement, a simple symbol, such as
the italic letter~$a$, to denote the same ratio or multiplier
which is more fully denoted by the complex symbol
$\displaystyle {{\sr b} \over {\sr a}}$.

It is an immediate consequence of these conceptions and
definitions, that the following relation holds good,
$${\nu \times {\sr a} \over \mu \times {\sr a}}
   =  {\nu \over \mu},
   \eqno (165.)$$
${\sr a}$ denoting any effective step, and $\mu$ and $\nu$
denoting any positive or contra-positive whole numbers; since the
fractional ratio denoted by the symbol
$\displaystyle {\nu \over \mu}$
is the ratio of the multiple step $\nu \times {\sr a}$ to the
multiple step $\mu \times {\sr a}$.  In like manner it follows,
from the same conceptions and definitions, that
$${\displaystyle {\nu \over \mu} \times {\sr b} \over {\sr b}}
   =  {\nu \over \mu},
   \quad\hbox{and reciprocally}\quad
  {\sr b}' = {{\nu \over \mu} \times {\sr b}}
   \quad\hbox{if}\quad
  {{\sr b}' \over {\sr b}} = {\nu \over \mu};
   \eqno (166.)$$
and more generally, that
$${\displaystyle {{\sr b} \over {\sr a}} \times {\sr c} \over {\sr c}}
   =  {{\sr b} \over {\sr a}},
   \eqno (167.)$$
and reciprocally,
$${\sr d} = {{\sr b} \over {\sr a}} \times {\sr c}
   \quad\hbox{if}\quad
  {{\sr d} \over {\sr c}} = {{\sr b} \over {\sr a}};
   \eqno (168.)$$
whatever effective steps may be denoted by
${\sr a}$, ${\sr b}$, ${\sr c}$, ${\sr d}$,
and whatever fraction by
$\displaystyle {\nu \over \mu}$.

We may also conceive combinations of ratios with each other, by
operations which we may call Addition, Subtraction,
Multiplication, and Division of Ratios, or of {\it general
algebraic numbers}, from the analogy of these operations to those
which we have already called by the same names, in the theories
of whole numbers and of fractions.  And as we wrote, in treating
of whole numbers,
$$\omega = \nu + \mu
   \quad\hbox{when}\quad
  \omega \times {\sr a} = (\nu \times {\sr a}) + (\mu \times {\sr a}),
   \eqno (107.)$$
and
$$\omega = \nu \times \mu
   \quad\hbox{when}\quad
  \omega \times {\sr a} = \nu \times (\mu \times {\sr a});
   \eqno (111.)$$
and, in the theory of fractions,
$${\nu'' \over \mu''} = {\nu' \over \mu'} + {\nu \over \mu}
   \quad\hbox{when}\quad
  {\nu'' \over \mu''} \times {\sr b}
      =    \left( {\nu' \over \mu'} \times {\sr b} \right)
         + \left( {\nu  \over \mu } \times {\sr b} \right),
   \eqno (149.)$$
and
$${\nu'' \over \mu''} = {\nu' \over \mu'} \times {\nu \over \mu}
   \quad\hbox{when}\quad
  {\nu'' \over \mu''} \times {\sr b}
      =  {\nu' \over \mu'} \times
            \left( {\nu \over \mu} \times {\sr b} \right),
   \eqno (152.)$$
with other similar expressions; so we shall now write, in the
more general theory of ratios,
$${{\sr b}'' \over {\sr a}''}
   =  {{\sr b}' \over {\sr a}'} + {{\sr b} \over {\sr a}}
   \quad\hbox{when}\quad
  {{\sr b}'' \over {\sr a}''} \times {\sr c}
   =     \left( {{\sr b}' \over {\sr a}'} \times {\sr c} \right)
       + \left( {{\sr b} \over {\sr a}}   \times {\sr c} \right),
   \eqno (169.)$$
and
$${{\sr b}'' \over {\sr a}''}
   =  {{\sr b}' \over {\sr a}'} \times {{\sr b} \over {\sr a}}
   \quad\hbox{when}\quad
  {{\sr b}'' \over {\sr a}''} \times {\sr c}
   =            {{\sr b}' \over {\sr a}'}
       \times \left( {{\sr b} \over {\sr a}}   \times {\sr c} \right):
   \eqno (170.)$$
and shall suppose that similar definitions are established for
the algebraical sums and products of more than two ratios, or
general algebraic numbers.  It follows that
$$\left. \eqalign{
{{\sr b}' \over {\sr a}} + {{\sr b} \over {\sr a}}
   &= {{\sr b}' + {\sr b} \over {\sr a}},\cr
{{\sr b}'' \over {\sr a}} + {{\sr b}' \over {\sr a}} + {{\sr b} \over {\sr a}}
   &= {{\sr b}'' + {\sr b}' + {\sr b} \over {\sr a}},\cr
\hbox{\&c.}\cr}
   \right\}
   \eqno (171.)$$
and that
$$\left. \eqalign{
{{\sr b}' \over {\sr b}} \times {{\sr b} \over {\sr a}}
   &= {{\sr b}' \over {\sr a}},\cr
{{\sr b}'' \over {\sr b}'} \times {{\sr b}' \over {\sr b}}
            \times {{\sr b} \over {\sr a}}
   &= {{\sr b}'' \over {\sr a}},
      \quad\hbox{\&c.}\cr}
   \right\}
   \eqno (172.)$$
A ratio between any two effective steps may be said to be
{\it positive\/} or {\it contra-positive}, according as those two
steps are {\it co-directional\/} or {\it contra-directional},
that is, according as their directions agree or differ; and then
the product of any two or more positive or contra-positive ratios
will evidently be contra-positive or positive according as there
are or are not an odd number of contra-positive ratios, as
factors of this product; because the direction of a step is not
altered or is restored, if it either be not reversed at all, or
be reversed an even number of times.

Again, we may say, as in the case of fractions, that we
{\it subtract\/} a ratio when we add its {\it opposite}, and that
we {\it divide\/} by a ratio when we multiply by its
{\it reciprocal}, if we agree to say that two ratios or numbers
are {\it opposites\/} when they generate {\it opposite steps\/}
by multiplication from one common step as a multiplicand, and if
we call them {\it reciprocals\/} when their corresponding acts of
multiplication are {\it opposite acts}, which destroy, each, the
effect of the other; and we may mark such opposites and
reciprocals, by writing, as in the notation of fractions,
$${{\sr b}' \over {\sr a}'} = \oppos {{\sr b} \over {\sr a}}
   \quad\hbox{when}\quad
  {{\sr b}' \over {\sr a}'} \times {\sr c}
   = \oppos \left( {{\sr b} \over {\sr a}} \times {\sr c} \right),
   \eqno (173.)$$
and
$${{\sr b}' \over {\sr a}'} = \recip {{\sr b} \over {\sr a}}
   \quad\hbox{when}\quad
  {{\sr b}' \over {\sr a}'} \times
         \left( {{\sr b} \over {\sr a}} \times {\sr c} \right)
   = {\sr c}:
   \eqno (174.)$$
definitions from which it follows that
$${\oppos {\sr b} \over {\sr a}}
   = \oppos {{\sr b} \over {\sr a}},
   \eqno (175.)$$
and that
$${{\sr a} \over {\sr b}} = \recip {{\sr b} \over {\sr a}}.
   \eqno (176.)$$

And as, by our conceptions and notations respecting the ordinal
relation of one fractional number to another, (as subsequent, or
coincident, or precedent, in the general progression of such
numbers from contra-positive to positive,) we had the relations,
$${\nu' \over \mu'} \gteqlt {\nu \over \mu},
   \quad\hbox{when}\quad
  {\nu' \over \mu'} \times {\sr a}
   \gteqlt {\nu \over \mu} \times {\sr a},\quad
      {\sr a} > 0;$$
so we may now establish, by analogous conceptions and notations
respecting ratios, the relations,
$${{\sr b}'' \over {\sr a}''} \gteqlt {{\sr b}' \over {\sr a}'},
   \quad\hbox{when}\quad
  {{\sr b}'' \over {\sr a}''} \times {\sr a}
      \gteqlt {{\sr b}' \over {\sr a}'} \times {\sr a},\quad
      {\sr a} > 0:
   \eqno (177.)$$
that is, more fully,
$${{\sr b}'' \over {\sr a}''} > {{\sr b}' \over {\sr a}'},
   \quad\hbox{when}\quad
   \left( {{\sr b}'' \over {\sr a}''} \times {\sr a} \right) + {\sc a}
      > \left( {{\sr b}' \over {\sr a}'} \times {\sr a} \right) + {\sc a},
   \eqno (178.)$$
$${{\sr b}'' \over {\sr a}''} = {{\sr b}' \over {\sr a}'},
   \quad\hbox{when}\quad
   \left( {{\sr b}'' \over {\sr a}''} \times {\sr a} \right) + {\sc a}
      = \left( {{\sr b}' \over {\sr a}'} \times {\sr a} \right) + {\sc a},
   \eqno (179.)$$
and
$${{\sr b}'' \over {\sr a}''} < {{\sr b}' \over {\sr a}'},
   \quad\hbox{when}\quad
   \left( {{\sr b}'' \over {\sr a}''} \times {\sr a} \right) + {\sc a}
      < \left( {{\sr b}' \over {\sr a}'} \times {\sr a} \right) + {\sc a};
   \eqno (180.)$$
the symbol~${\sc a}$ denoting any moment of time, and ${\sr a}$
any late-making step.  The relation (179.) is indeed an immediate
consequence of the first conceptions of steps and ratios; but it
is inserted here along with the relations (178.) and (180.), to
show more distinctly in what manner the comparison and
arrangement of the moments
$$\left( {{\sr b}'  \over {\sr a}' } \times {\sr a} \right) + {\sc a},\quad
  \left( {{\sr b}'' \over {\sr a}''} \times {\sr a} \right) + {\sc a},
   \quad\hbox{\&c.}
   \eqno (181.)$$
which are suggested and determined by the ratios or numbers
$\displaystyle {{\sr b}'  \over {\sr a}'}$,
$\displaystyle {{\sr b}'' \over {\sr a}''}$, \&c.,
(in combination with a standard moment~${\sc a}$ and with a
late-making step~${\sr a}$,) enable us to compare and arrange
those ratios or numbers themselves, and to conceive an indefinite
progression of ratio from contra-positive to positive, including
the indefinite progression of whole numbers (103.), and the more
comprehensive progression of fractional numbers considered in the
17th article: for it will soon be shown, that though every
fractional number is a ratio, yet there are many ratios which
cannot be expressed under the form of fractional numbers.
Meanwhile we may observe, that the theorems (151.) (157.) (158.)
respecting the ordinal relations of fractions in the general
progression of number, are true, even when the symbols
$\alpha$~$\beta$~$\gamma$ denote ratios which are not reducible
to the fractional form; and that this indefinite progression of
number, or of ratio, from contra-positive to positive,
corresponds in all respects to the thought from which it was
deduced, of the progression of time itself, from moments
indefinitely early to moments indefinitely late.

\bigbreak

22.
It is manifest, on a little attention, that the ratio of one
effective step~${\sr b}$ to another~${\sr a}$, is a relation
which is entirely determined when those steps are given, but
which is not altered by multiplying both those steps by any
common multiplier, whether positive or contra-positive; for the
{\it relative largeness\/} of the two steps is not altered by
doubling or halving both, or by enlarging or diminishing the
magnitudes of both in any other common ratio of magnitude, that
is, by multiplying both by any common positive multiplier: nor is
their {\it relative direction\/} altered, by reversing the
directions of both.  We have then, generally,
$${\displaystyle {{\sr b}' \over {\sr a}'} \times {\sr b} \over
      \displaystyle {{\sr b}' \over {\sr a}'} \times {\sr a}}
   =  {{\sr b} \over {\sr a}};
   \eqno (182.)$$
and in particular, by changing ${\sr a}'$ to ${\sr a}$, and
${\sr b}'$ to ${\sr c}$,
$${\displaystyle {{\sr c} \over {\sr a}} \times {\sr b} \over {\sr c}}
   =  {{\sr b} \over {\sr a}}.
   \eqno (183.)$$
Hence, by (167.), the two steps
$\displaystyle {{\sr c} \over {\sr a}} \times {\sr b}$ and
$\displaystyle {{\sr b} \over {\sr a}} \times {\sr c}$
are related in one common ratio, namely the ratio
$\displaystyle {{\sr b} \over {\sr a}}$,
to the common step~${\sr c}$, and therefore are equivalent to
each other; that is, we have the equation,
$${{\sr c} \over {\sr a}} \times {\sr b}
   = {{\sr b} \over {\sr a}} \times {\sr c},
   \eqno (184.)$$
whatever three effective steps may be denoted by
${\sr a}$~${\sr b}$~${\sr c}$.

In general, when any four effective steps
${\sr a}$~${\sr b}$~${\sr c}$~${\sr d}$
are connected by the relation
$${{\sr d} \over {\sr c}} = {{\sr b} \over {\sr a}},
   \eqno (185.)$$
that is,  when the ratio of the step~${\sr d}$ to ${\sr c}$ is
the same as the ratio of the step~${\sr b}$ to ${\sr a}$, these
two pairs of steps ${\sr a}$, ${\sr b}$ and ${\sr c}$, ${\sr d}$
may be said to be {\it analogous\/} or {\it proportional
pairs\/}; the steps ${\sr a}$ and ${\sr c}$ being called the
{\it antecedents\/} of the analogy, (or of the proportion) and
the steps ${\sr b}$ and ${\sr d}$ being called the
{\it consequents}, while ${\sr a}$ and ${\sr d}$ are the
{\it extremes\/} and ${\sr b}$ and ${\sr c}$ the {\it means}.
And since the last of these four steps, or the second
consequent~${\sr d}$, may, by (168.), be expresssed by the symbol
$\displaystyle {{\sr b} \over {\sr a}} \times {\sr c}$,
we see, by (184.), that it bears to the first
consequent~${\sr b}$ the ratio
$\displaystyle {{\sr c} \over {\sr a}}$
of the second antecedent~${\sr c}$ to the first
antecedent~${\sr a}$; that is,
$${{\sr d} \over {\sr b}} = {{\sr c} \over {\sr a}}
   \quad\hbox{if}\quad
  {{\sr d} \over {\sr c}} = {{\sr b} \over {\sr a}}:
   \eqno (186.)$$
a theorem which shows that we may transform the expression of an
{\it analogy\/} (or {\it proportion\/}) between two pairs of
effective steps in a manner which may be called
{\it alternation}.  (Compare the theorem (11.).)

It is still more easy to perceive that we may {\it invert\/} an
analogy or proportion between any two pairs of effective steps;
or that the following theorem is true,
$${{\sr c} \over {\sr d}} = {{\sr a} \over {\sr b}},
   \quad\hbox{if}\quad
  {{\sr d} \over {\sr c}} = {{\sr b} \over {\sr a}}.
   \eqno (187.)$$
Combining inversion with alternation, we see that
$${{\sr b} \over {\sr d}} = {{\sr a} \over {\sr c}},
   \quad\hbox{if}\quad
  {{\sr d} \over {\sr c}} = {{\sr b} \over {\sr a}}.
   \eqno (188.)$$
(Compare the theorems (12.) and (13.).)

In general, if any two pairs of effective steps
${\sr a}$, ${\sr b}$ and ${\sr c}$, ${\sr d}$ be analogous, we do
not disturb this analogy by interchanging the extremes among
themselves, or the means among themselves, or by substituting
extremes for means and means for extremes; or by altering
{\it proportionally}, that is, altering in one common ratio, or
multiplying by one common multiplier, whether positive or
contra-positive, the two consequents, or the two antecedents, or
the two steps of either pair: or, finally, by altering {\it in
inverse proportion}, that is, multiplying respectively by any two
reciprocal multipliers, the two extremes, or the two means.  The
analogy (185.) may therefore be expressed, not only in the ways
(186.), (187.), (188.), but also in the following:
$${a \times {\sr d} \over {\sr c}}
   =  {a \times {\sr b} \over {\sr a}},\quad
  {{\sr d} \over a \times{\sr c}}
   =  {{\sr b} \over a \times {\sr a}},\quad
  {a \times {\sr d} \over a \times {\sr c}}
   =  {{\sr b} \over {\sr a}},
   \eqno (189.)$$
$${\recip a \times {\sr d} \over {\sr c}}
   =  {{\sr b} \over a \times {\sr a}},\quad
  {{\sr d} \over \recip a \times {\sr c}}
  =  {a \times {\sr b} \over {\sr a}},
   \eqno (190.)$$
$a$ denoting any ratio of one effective step to another, and
$\recip a$ denoting the reciprocal ratio, of the latter step to
the former.

\bigbreak

23.
We may also consider it as evident that if any effective
step~${\sr c}$ be compounded of any others ${\sr a}$ and
${\sr b}$, this relation of compound and components will not be
disturbed by altering the magnitudes of all in any common ratio
of magnitude, that is by doubling or halving it, or multiplying
all by any common positive multiplier; and we saw, in the 12th
article, that the same relation of compound and components is not
disturbed by reversing the directions of all: we may therefore
multiply all by any common multiplier~$a$, whether positive or
contra-positive, and may establish the theorem,
$$a \times {\sr c} = (a \times {\sr b}) + (a \times {\sr a}),
   \quad\hbox{if}\quad
  {\sr c} = {\sr b} + {\sr a};
   \eqno (191.)$$
which gives, by the definitions (169.) (170.) for the sum and
product of two ratios, this other important relation,
$$a \times (b' + b) = (a \times b') + (a \times b),
   \eqno (192.)$$
if $b$, $b'$, and $b' + b$, denote any three positive or
contra-positive numbers, connected with each other by the
definition (169.), or by the following condition,
$$(b' + b) \times {\sr d}
   =  (b' \times {\sr d}) + (b \times {\sr d}),
   \eqno (193.)$$
in which ${\sr d}$ denotes any arbitrary effective step.  The
definitions of the sum and product of two ratios, or algebraic
numbers, give still more simply the theorem,
$$(b' + b) \times a = (b' \times a) + (b \times a).
   \eqno (194.)$$

The definition (169.) of a sum of two ratios, when combined with
the theorem (75.) respecting the arbitrary order of composition
of two successive steps, gives the following similar theorem
respecting the addition of two ratios,
$$b + a = a + b.
   \eqno (195.)$$
And if the definition (170.) of a product of two ratios or
multipliers be combined with the theorem (186.) of alternation of
an analogy between two pairs of steps, in the same way as the
definition of a compound step was combined in the 12th article
with the theorem of alternation of an analogy between two pairs
of moments, it shows that as any two steps ${\sr a}$, ${\sr b}$,
may be applied to any moment, or compounded with each other,
either in one or in the opposite order,
(${\sr b} + {\sr a} = {\sr a} + {\sr b}$,)
so any two ratios $a$ and $b$ may be applied as multipliers to
any step, or combined as factors of a product with each other, in
an equally arbitrary order; that is, we have the relation,
$$b \times a = a \times b.
   \eqno (196.)$$
It is easy to infer, from the theorems (195.) (196.), that the
opposite of a sum of two ratios is the sum of the opposites of
those ratios, and that the reciprocal of the product of two
ratios is the product of their two reciprocals; that is,
$$\oppos (b + a) = \oppos b + \oppos a,
   \eqno (197.)$$
and
$$\recip (b \times a) = \recip b \times \recip a.
   \eqno (198.)$$
And all the theorems of this article, respecting pairs of ratios
or of steps, may easily be extended to the comparison and
combination of more ratios or steps than two.  In particular,
when any number of ratios are to be added or multiplied together,
we may arrange them in any arbitrary order; and in any
multiplication of ratios, we may treat any one factor as the
algebraic sum of any number of other ratios, or partial factors,
and substitute each of these separately and successively for it,
and the sum of the partial products thus obtained will be the
total product sought.  As an example of the multiplication of
ratios, considered thus as sums, it is plain from the foregoing
principles that
$$\eqalignno{
(d + c) \times (b + a)
   &= \{ d \times (b + a) \} + \{ c \times (b + a) \} \cr
   &= (d \times b) + (d \times a) + (c \times b) + (c \times a) \cr
   &= db + da + cb + ca,
      &(199.)\cr}$$
and that
$$\eqalignno{
(b + a) \times (b + a)
   &= (b \times b) + (2 \times b \times a) + (a \times a) \cr
   &= bb + 2ba + aa,
      &(200.)\cr}$$
whatever positive or contra-positive ratios may be denoted by
$a$~$b$~$c$~$d$.

And though we have only considered effective steps, and positive
or contra-positive ratios, (or algebraic numbers,) in the few
last articles of this Essay, yet the results extend to null
steps, and to null ratios, also; provided that for the reasons
given in the 20th article we treat all such null steps as
consequents only and not as antecedents of ratios, admitting null
ratios themselves but not their reciprocals into our formul{\ae},
or employing null numbers as multipliers only but not as
divisors, in order to avoid the introduction of symbols which
suggest either impossible or indeterminate operations.

\bigbreak

\centerline{\it
On the insertion of a Mean Proportional between two steps; and on
Impossible, Ambiguous,}

\nobreak\vskip 3pt

\centerline{\it
and Incommensurable Square-Roots of Ratios.}

\nobreak\bigskip

24.
Three effective steps ${\sr a}$~${\sr b}$~${\sr b}'$ may be said
to form a {\it continued analogy\/} or {\it continued
proportion}, when the ratio of ${\sr b}'$ to ${\sr b}$ is the
same as that of ${\sr b}$ to ${\sr a}$, that is, when
$${{\sr b}' \over {\sr b}} = {{\sr b} \over {\sr a}};
   \eqno (201.)$$
${\sr a}$ and ${\sr a}'$ being then the {\it extremes}, and
${\sr b}$ the {\it mean}, or the {\it mean proportional\/}
between ${\sr a}$ and ${\sr b}'$, in this continued analogy; in
which ${\sr b}'$ is also the {\it third proportional\/} to
${\sr a}$ and ${\sr b}$, and ${\sr a}$ is at the same time the
third proportional to ${\sr b}'$ and ${\sr b}$, because the
analogy may be inverted thus,
$${{\sr a} \over {\sr b}} = {{\sr b} \over {\sr b}'}.
   \eqno (202.)$$
When the condition (201.) is satisfied, we may express $b'$ as
follows,
$${\sr b}' = {{\sr b} \over {\sr a}} \times {\sr b};
   \eqno (203.)$$
that is, if we denote by $a$ the ratio of ${\sr b}$ to ${\sr a}$,
we shall have the relations
$${\sr b} = a \times {\sr a},\quad
  {\sr b}' = a \times {\sr b} = a \times a \times {\sr a};
   \eqno (204.)$$
and reciprocally when these relations exist, we can conclude the
existence of the continued analogy (201.).  It is clear that
whatever effective steps may be denoted by ${\sr a}$ and
${\sr b}$, we can always determine, (or conceive determined,) in
this manner, one third proportional~${\sr b}'$ and only one; that
is, we can complete the continued analogy (201.) in one, but in
only one way, when an extreme ${\sr a}$ and the mean~${\sr b}$
are given: and it is important to observe that whether the
ratio~$a$ of the given mean~${\sr b}$ to the given
extreme~${\sr a}$ be positive or contra-positive, that is,
whether the two given steps ${\sr a}$ and ${\sr b}$ be
co-directional or contra-directional steps, the product
$a \times a$ will necessarily be a positive ratio, and therefore
the deduced extreme step~${\sr b}'$ will necessarily be
co-directional with the given extreme step~${\sr a}$.  In fact,
without recurring to the theorem of the 21st article respecting
the cases in which a product of contra-positive factors is
positive, it is plain that the continued analogy requires, by its
conception, that the step~${\sr b}'$ should be co-directional to
${\sr b}$, if ${\sr b}$ be co-directional to ${\sr a}$, and that
${\sr b}'$ should be contra-directional to ${\sr b}$ if ${\sr b}$
be contra-directional to ${\sr a}$; so that in every possible
case the extremes themselves are co-directional, as both agreeing
with the mean or both differing from the mean in direction.
{\it It is, therefore, impossible to insert a mean proportional
between two contra-directional steps\/}; but for the same reason
{\it we may insert either of two opposite steps as a mean
proportional between two given co-directional steps\/}; namely,
either a step which agrees with each, or a step which differs
from each in direction, while the common magnitude of these two
opposite steps is exactly intermediate in the way of ratio
between the magnitudes of the two given extremes.  (We here
assume, as it seems reasonable to do, the conception of the
general existence of such an exactly intermediate magnitude,
although the nature and necessity of this conception will soon be
more fully considered.)  For example, it is impossible to insert
a mean proportional between the two contra-directional
(effective) steps ${\sr a}$ and $\oppos 9 {\sr a}$, that is, it
is impossible to find any step~${\sr b}$ which shall satisfy the
conditions of the continued analogy
$${\oppos 9 {\sr a} \over {\sr b}} = {{\sr b} \over {\sr a}},
   \eqno (205.)$$
or any number or ratio~$a$ which shall satisfy the equation
$$a \times a = \oppos 9:
   \eqno (206.)$$
whereas it is possible to insert in two different ways a mean
proportional~${\sr b}$ between the two co-directional (effective)
steps ${\sr a}$ and $9 {\sr a}$, or to satisfy by two different
steps ${\sr b}$ (namely, by the step $3 {\sr a}$, and also by the
opposite step $\oppos 3 {\sr a}$) the conditions of the continued
analogy
$${9 {\sr a} \over {\sr b}} = {{\sr b} \over {\sr a}},
   \eqno (207.)$$
and it is possible to satisfy by two different ratios~$a$ the
equation
$$a \times a = 9,
   \eqno (208.)$$
namely, either by the ratio~$3$ or by the opposite ratio
$\oppos 3$.  In general, we may agree to express the two opposite
ratios~$a$ which satisfy the equation
$$a \times a = b \, (> 0),
   \eqno (209.)$$
by the two symbols
$$\surd b \, (> 0)
   \quad\hbox{and}\quad
  \oppos \surd b \, (< 0),
   \eqno (210.)$$
$b$ and $\surd b$ being positive ratios, but $\oppos \surd b$
being contra-positive; for example,
$$\surd 9 = 3,\quad \oppos \surd 9 = \oppos 3.
   \eqno (211.)$$
With this notation we may represent the two opposite steps of
which each is a mean proportional between two given
co-directional (effective) steps ${\sr a}$ and ${\sr b}'$, by the
symbols
$$\sqrt{ {{\sr b}' \over {\sr a}} } \times {\sr a},
   \quad\hbox{and}\quad
  \oppos \sqrt{ {{\sr b}' \over {\sr a}} } \times {\sr a};
   \eqno (212.)$$
and shall have for each the equation of a continued analogy,
$${{\sr b}' \over \displaystyle
         \sqrt{ {{\sr b}' \over {\sr a}} } \times {\sr a}}
   =  {\displaystyle \sqrt{ {{\sr b}' \over {\sr a}} } \times {\sr a}
         \over {\sr a}},\quad
  {{\sr b}' \over \displaystyle
         \oppos \sqrt{ {{\sr b}' \over {\sr a}} } \times {\sr a}}
   =  {\displaystyle \oppos \sqrt{ {{\sr b}' \over {\sr a}} } \times {\sr a}
         \over {\sr a}}.
   \eqno (213.)$$
We may also call the numbers $\surd b$ and $\oppos \surd b$ by
the common name of {\it roots}, or (more fully)
{\it square-roots\/} of the positive number~$b$; distinguishing
them from each other by the separate names of the {\it positive
square-root\/} and the {\it contra-positive square-root\/} of
that number~$b$, which may be called their common {\it square\/}:
though we may sometimes speak simply of {\it the square-root\/}
of a (positive) number, meaning then the positive root, which is
simpler and more important than the other.

\bigbreak

25.
The idea of the {\it continuity of the progression from moment to
moment in time\/} involves the idea of a similarly
{\it continuous progression in magnitude\/} from any one
effective step or interval between two different moments, to any
other unequal effective step or other unequal interval; and also
the idea of a {\it continuous progression in ratio}, from any one
degree of inequality, in the way of relative largeness or
smallness, as a relation between two steps, to any other degree.
Pursuing this train of thought, we find ourselves compelled to
conceive the existence (assumed in the last article) of a
determined magnitude~${\sr b}$, exactly intermediate in the way
of ratio between any two given unequal magnitudes ${\sr a}$ and
${\sr b}'$, that is, larger or smaller than the one, in exactly
the same proportion in which it is smaller or larger than the
other; and therefore also the existence of a determined number or
ratio~${\sr a}$ which is the exact square-root of any proposed
(positive) number or ratio~$b$.  To show this more fully, let
${\sc a}$~${\sc b}$~${\sc d}$ be any three given distinct
moments, connected by the relations
$${{\sc d} - {\sc a} \over {\sc b} - {\sc a}} = b,\quad b > 1,
   \eqno (214.)$$
which require that the moment~${\sc b}$ should be situated
between ${\sc a}$ and ${\sc d}$; and let ${\sc c}$ be any fourth
moment, lying between ${\sc b}$ and ${\sc d}$, but capable of
being chosen as near to ${\sc b}$ or as near to ${\sc d}$ as we
may desire, in the continuous progression of time.  Then the two
ratios
$${{\sc c} - {\sc a} \over {\sc b} - {\sc a}}
   \quad\hbox{and}\quad
  {{\sc d} - {\sc a} \over {\sc c} - {\sc a}}$$
will both be positive ratios, and both will be {\it ratios of
largeness}, (that is, each will be a ratio of a larger to a
smaller step,) which we may denote for abridgement as follows,
$${{\sc c} - {\sc a} \over {\sc b} - {\sc a}} = x,\quad
  {{\sc d} - {\sc a} \over {\sc b} - {\sc a}} = y = \recip x \times b;
   \eqno (215.)$$
but by choosing the moment~${\sc c}$ sufficiently near to
${\sc b}$ we may make the ratio~$x$ approach as near as we desire
to the ratio of equality denoted by $1$ while the ratio~$y$ will
tend to the given ratio of largeness denoted by $b$; results
which we may express by the following written sentence,
$$\hbox{if}\quad
   \underline{\rm L} {\sc c} = {\sc b},
   \quad\hbox{then}\quad
   \underline{\rm L} x = 1
   \quad\hbox{and}\quad
   \underline{\rm L} y = b,
   \eqno (216.)$$
prefixing the symbol~$\underline{\rm L}$, (namely the letter~$L$
of the Latin word Limes, distinguished by a bar drawn under it,)
to the respective marks of the variable moment~${\sc c}$ and
variable ratios $x$,~$y$, in order to denote the respective
{\it limits\/} to which those variables tend, while we vary the
selection of one of them, and therefore also of the rest.  Again,
we may choose the moment~${\sc c}$ nearer and nearer to
${\sc d}$, and then the ratio~$x$ will tend to the given ratio of
largeness denoted by $b$, while the ratio~$y$ will tend to the
ratio of equality; that is,
$$\hbox{if}\quad
   \underline{\rm L} {\sc c} = {\sc d},
   \quad\hbox{then}\quad
   \underline{\rm L} x = b
   \quad\hbox{and}\quad
   \underline{\rm L} y = 1;
   \eqno (217.)$$
and if we conceive a continuous progression of moments ${\sc c}$
from ${\sc b}$ to ${\sc d}$, we shall also have a continuous
progression of ratios~$x$, determining higher and higher degrees
of relative largeness (of the increasing step ${\sc c} - {\sc a}$
as compared with the fixed step ${\sc b} - {\sc a}$) from the
ratio of equality~$1$ to the given ratio of largeness~$b$,
together with another continuous but opposite progression of
ratios~$y$, determining lower and lower degrees of relative
largeness (of the fixed step ${\sc d} - {\sc a}$ as compared with
the increasing step ${\sc c} - {\sc a}$) from the same given
ratio of largeness $b$ down to the ratio of equality~$1$; so that
we cannot avoid conceiving the existence of some one determined
state of the progression of the moment~${\sc c}$, for which the
two progressions of ratio {\it meet}, and for which they give
$$\recip x \times b = y = x,
   \quad\hbox{that is}\quad
  {{\sc d} - {\sc a} \over {\sc c} - {\sc a}}
   = {{\sc c} - {\sc a} \over {\sc b} - {\sc a}},
   \eqno (218.)$$
having given at first $y > x$, and giving afterwards $y < x$.
And since, in general,
$${{\sc d} - {\sc a} \over {\sc c} - {\sc a}}
      \times {{\sc c} - {\sc a} \over {\sc b} - {\sc a}}
   =  {{\sc d} - {\sc a} \over {\sc b} - {\sc a}},
   \quad\hbox{that is,}\quad
  (\recip x \times b) \times x = b,
   \eqno (219.)$$
we can and must by (218.) and (214.), conceive the existence of a
positive ratio~$a$ which shall satisfy the condition (209.),
$a \times a = b$, if $b > 1$, that is, we must conceive the
existence of a positive square-root of $b$, if $b$ denote any
positive ratio of largeness.  A reasoning of an entirely similar
kind would prove that we must conceive the existence of a
positive square-root of $b$, when $b$ denotes any positive ratio
of smallness, ($b < 1$;) and if $b$ denote the positive ratio of
equality, ($b = 1$,) then it evidently has that ratio of equality
itself for a positive square-root.  We see then by this more full
examination what we before assumed to be true, that every
positive number or ratio~$b$ has a positive (and therefore also a
contra-positive) square-root.

And hence we can easily prove another important property of
ratios, which has been already mentioned without proof; namely
that several ratios can and must be conceived to exist, which are
incapable of being expressed under the form of whole or
fractional numbers; or, in other words, that every effective
step~${\sr a}$ has other steps {\it incommensurable\/} with it;
and therefore that when any two distinct moments ${\sc a}$ and
${\sc b}$ are given, it is possible to assign (in various ways) a
third moment~${\sc c}$ which shall not be {\it uniserial\/} with
these two, in the sense of the 8th article, that is, shall not
belong in common with them to any one equi-distant series of
moments, comprising all the three.  For example, the positive
square-root of $2$, which is evidently intermediate between $1$
and $2$ in the general progression of numbers, and which
therefore is not a whole number, cannot be expressed as a
fractional number either; since if it could be put under the
fractional form $\displaystyle {n \over m}$, so that
$$\sqrt{2} = {n \over m},
   \eqno (220.)$$
we should then have
$$2 = {n \over m} \times {n \over m}
    = {n \times n \over m \times m},
   \eqno (221.)$$
that is,
$$n \times n = 2 \times m \times m;
   \eqno (222.)$$
but the arithmetical properties of quotities are sufficient to
prove that this last equation is impossible, whatever positive
whole numbers may be denoted by $m$ and $n$.  And hence, if we
imagine that
$${\sr b} = \sqrt{2} \times {\sr a},\quad {\sr a} > 0,
   \eqno (223.)$$
the step~${\sr b}$ which is a mean proportional between the two
effective and co-directional steps ${\sr a}$ and $2 {\sr a}$ (of
which the latter is double the former) will be
{\it incommensurable\/} with the step~${\sr a}$ (and therefore
also with the double step $2 {\sr a}$); that is, we cannot find
nor conceive any other step~${\sr c}$ which shall be a
{\it common measurer\/} of the steps ${\sr a}$ and ${\sr b}$, so
as to satisfy the conditions
$${\sr a} = m {\sr c},\quad
  {\sr b} = n {\sr c},
   \eqno (224.)$$
whatever positive or contra-positive whole numbers we may denote
by $m$ and $n$; because, if we could do this, we should then have
the relations,
$${\sr b} = {n \over m} {\sr a},\quad
  \sqrt{2} = {n \over m},
   \eqno (225.)$$
of which the latter has been shown to be impossible.  Hence
finally, if ${\sc a}$ and ${\sc b}$ be any two distinct moments,
and if we choose a third moment~${\sc c}$ such that
$${{\sc c} - {\sc a} \over {\sc b} - {\sc a}} = \surd 2,
   \eqno (226.)$$
the moment~${\sc c}$ will not be uniserial with ${\sc a}$ and
${\sc b}$, that is, no one equi-distant series of moments can be
imagined, comprising all the three.  And all that has here been
shown respecting the square-root of two, extends to the
square-root of three, and may be illustrated and applied in an
infinite variety of other examples.  We must then admit the
existence of pairs of steps which have no common measurer; and
may call the ratio between any two such steps an
{\it incommensurable ratio}, or {\it incommensurable number}.

\bigbreak

\centerline{\it
More formal proof of the general existence of a determined
positive square-root,}

\nobreak\vskip 3pt

\centerline{\it
commensurable or incommensurable, for every determined positive
ratio: continuity of}

\nobreak\vskip 3pt

\centerline{\it
progression of the square, and principles connected with this continuity.}

\nobreak\bigskip

26.
The existence of these incommensurables, (or the necessity of
conceiving them to exist,) is so curious and remarkable a result,
that it may be usefully confirmed by an additional proof of the
general existence of square-roots of positive ratios, which will
also offer an opportunity of considering some other important
principles.

The existence of a positive square-root $a = \surd b$, of any
proposed ratio of largeness $b > 1$, was proved in the foregoing
article, by the comparison of the two opposite progressions of
the two ratios $x$ and $\recip x \times b$, from the states
$x = 1$, $\recip x \times b = b$, for which
$\recip x \times b > x$, to the states
$x = b$, $\recip x \times b = 1$, for which
$\recip x \times b < x$; for this comparison obliged us to
conceive the existence of an intermediate state or ratio~$a$
between the limits $1$ and $b$, as a {\it common state\/} or
{\it state of meeting\/} of these two opposite progressions,
corresponding to the conception of a {\it moment\/} at which the
decreasing ratio $\recip x \times b$ {\it becomes exactly
equal\/} to the increasing ratio~$x$, having been {\it previously
a greater ratio\/} (or a ratio of greater relative largeness
between steps), and becoming {\it afterwards a lesser ratio\/}
(or a ratio of less relative largeness).  And it was remarked
that an exactly similar comparison of two other inverse
progressions would prove the existence of a positive square-root
$\surd b$ of any proposed positive ratio~$b$ of smallness, $b <
1$, $b > 0$.  But instead of thus comparing, with the progression
of the positive ratio~$x$, the connected but opposite progression
of the connected positive ratio $\recip x \times b$, and showing
that these progressions meet each other in a certain intermediate
state or positive ratio~$a$, we might have compared the two
connected and not opposite progressions of the two connected
positive ratios $x$ and $x \times x$, of which the latter is the
square of the former; and might have shown that the square
($= x \times x = x x$) increases {\it constantly and
continuously\/} with the root ($= x$), from the state zero, so as
to {\it pass successively through every state\/} of positive
ratio~$b$.  To develope this last conception, and to draw from it
a more formal (if not a more convincing) proof than that already
given, of the necessary existence of a conceivable positive
square-root for every conceivable positive number, we shall here
lay down a few {\it Lemmas}, or preliminary and auxiliary
propositions.

{\it Lemma\/}~I.
$$\hbox{If}\quad
   x' \gteqlt x,
   \quad\hbox{and}\quad
   x > 0,\quad x' > 0,
   \quad\hbox{then}\quad
   x' x' \gteqlt x x;
   \eqno (227.)$$
that is, the square $x' x'$ of any one positive number or
ratio~$x'$ is greater than, or equal to, or less than the square
$x x$ of any other positive number or ratio~$x$, according as the
number~$x'$ itself is greater than, or equal to, or less than the
number~$x$; one number~$x'$ being said to be {\it greater\/} or
{\it less\/} than another number~$x$, when it is on the positive
or on the contra-positive side of that other, in the general
progression of numbers considered in the 21st article.  This
Lemma may be easily proved from the conceptions of ratios and of
squares; it follows also without difficulty from the theorem of
multiplication (200.).  And hence we may obviously deduce as a
{\it corollary\/} of the foregoing Lemma, this converse
proposition:
$$\hbox{if}\quad
   x' x' \gteqlt x x,
   \quad\hbox{and}\quad
   x > 0,\quad x' > 0,
   \quad\hbox{then}\quad
   x' \gteqlt x;
   \eqno (228.)$$
that is, if any two proposed positive numbers have positive
square-roots, the root of the one number is greater than, or
equal to, or less than the root of the other number, according as
the former proposed number itself is greater than, or equal to,
or less than the latter proposed number.

The foregoing Lemma shows that the square {\it constantly\/}
increases with the root, from zero up to states indefinitely
greater and greater.  But to show that this increase is
{\it continuous\/} as well as constant, and to make more distinct
the conception of such continuous increase, these other Lemmas
may be added.

{\it Lemma\/}~II.
If $a'$ and $a''$ be any two unequal ratios, we can and must
conceive the existence of some intermediate ratio~$a$; that is,
we can always choose $a$ or conceive it chosen so that
$$a > a',\quad a < a'',
   \quad\hbox{if}\quad a'' > a'.
   \eqno (229.)$$
For then we have the following relation of subsequence between
moments,
$$a'' ({\sc b} - {\sc a}) + {\sc a}
   > a' ({\sc b} - {\sc a}) + {\sc a},
   \quad\hbox{if}\quad
   {\sc b} > {\sc a},
   \eqno (230.)$$
by the very meaning of the relation of subsequence between
ratios, $a'' > a'$, as defined in article~21.; and between any
two distinct moments it is manifestly possible to insert an
intermediate moment, indeed as many such as we may desire: it is,
therefore, possible to insert a moment~${\sc c}$ between the two
non-coincident moments
$$a'  ({\sc b} - {\sc a}) + {\sc a}
   \quad\hbox{and}\quad
  a'' ({\sc b} - {\sc a}) + {\sc a},$$
such that
$${\sc c} > a'  ({\sc b} - {\sc a}) + {\sc a},\quad
  {\sc c} < a'' ({\sc b} - {\sc a}) + {\sc a},
   \quad\hbox{if}\quad
   {\sc b} > {\sc a},\quad a'' > a';
   \eqno (231.)$$
and then if we put, for abridgement,
$$a = {{\sc c} - {\sc a} \over {\sc b} - {\sc a}},
   \eqno (232.)$$
denoting by $a$ the ratio of the step or interval
${\sc c} - {\sc a}$ to the step or interval ${\sc b} - {\sc a}$,
we shall have
$$\left. \eqalign{
{\sc c} = a ({\sc b} - {\sc a}) + {\sc a},\quad {\sc b} > {\sc a},\cr
a ({\sc b} - {\sc a}) + {\sc a} > a'  ({\sc b} - {\sc a}) + {\sc a},\cr
a ({\sc b} - {\sc a}) + {\sc a} < a'' ({\sc b} - {\sc a}) + {\sc a},\cr}
   \right\}
   \eqno (233.)$$
and therefore finally,
$$a > a',\quad a < a'',$$
as was asserted in the Lemma.  We see, too, that the ratio~$a$ is
not determined by the conditions of that Lemma, but that an
indefinite variety of ratios may be chosen, which shall all
satisfy those conditions.

{\it Corollary}.
It is possible to choose, or conceive chosen, a ratio~$a$, which
shall satisfy all the following conditions,
$$\left. \eqalign{
a > a' ,\quad a > b' ,\quad a > c' ,\ldots \cr
a < a'',\quad a < b'',\quad a < c'',\ldots \cr}
   \right\}
   \eqno (234.)$$
if the least (or hindmost) of the ratios
$a'', b'', c'',\ldots$ be greater (or farther advanced in the
general progression of ratio from contra-positive to positive)
than the greatest (or foremost in that general progression) of
the ratios $a', b', c'$, \&c.

For if $c''$ (for example) be the least or hindmost of the ratios
$a'', b'', c'',\ldots$ so that
$$c'' \leq a'',\quad c'' \leq b'',\quad c'' \leq d'',\ldots
   \eqno (235.)$$
and if $b'$ (for example) be the greatest or foremost of the
ratios $a', b', c',\ldots$ so that
$$b' \geq a',\quad b' \geq c',\quad b' \geq d',\ldots
   \eqno (236.)$$
(the abridged sign $\leq$ denoting what might be more fully
written thus, ``$<$ or $=$'', and the other abridged sign~$\geq$
denoting in like manner ``$>$ or $=$'',) then the conditions
(234.) of the Corollary will all be satisfied, if we can satisfy
these two conditions,
$$a > b',\quad a < c'';
   \eqno (237.)$$
and this, by the Lemma, it is possible to do, if we have the
relation
$$c'' > b',
   \eqno (238.)$$
which relation the enunciation of the Corollary supposes to
exist.

{\it Remark}.---If the ratios
$a' \, b' \, c' \,\ldots \, a'' \, b'' \, c'' \,\ldots$
be all actually given, and therefore limited in number; or if,
more generally, the least of the ratios
$a'' \, b'' \, c'' \, \ldots$
and the greatest of the ratios $a' \, b' \, c' \, \ldots$ be
actually given and determined, so that we have only to choose a
ratio~$a$ intermediate between two given unequal ratios; we can
then make this choice in an indefinite variety of ways, even if
it should be farther required that $a$ should be a fractional
number $\displaystyle {\nu \over \mu}$, since we saw, in the 8th
article, that between any two distinct moments, such as
$a' ({\sc b} - {\sc a}) + {\sc a}$ and
$a'' ({\sc b} - {\sc a}) + {\sc a}$, it is possible to insert
an indefinite variety of others, such as
$\displaystyle {\nu \over \mu} ({\sc b} - {\sc a}) + {\sc a}$,
{\it uniserial\/} with the two moments ${\sc a}$ and ${\sc b}$,
and giving therefore fractions such as
$\displaystyle {\nu \over \mu}$,
intermediate (by the 21st article) between the ratios $a'$ and
$a''$.  But if, instead of actually knowing the ratios
$a' \, b' \, c' \, \ldots \, a'' \, b'' \, c'' \, \ldots$
themselves, in (234.), we only know a {\it law\/} by which we may
assign such ratios without end, this law may lead us to conceive
new conditions of the form (234.), incompatible with some (and
perhaps ultimately with all) of these selections of fractional
ratios
$\displaystyle {\nu \over \mu}$,
although they can never exclude {\it all ratios~$a$ whatever},
unless they be incompatible with each other, that is, unless they
fail to possess the relation mentioned in the Corollary.  The
force of this remark will soon be felt more fully.

{\it Lemma\/}~III.
If $b$ denote any given positive ratio, whether it be or be not
the square of any whole or any fractional number, it is possible
to find, or to conceive as found, one positive ratio~$a$, and
only one, which shall satisfy all the conditions of the following
forms:
$$a > {n' \over m'},\quad a < {n'' \over m''},
   \eqno (239.)$$
$m' \, n' \, m'' \, n''$ denoting here any positive whole numbers
whatever, which can be chosen so as to satisfy these relations,
$${n' n' \over m' m'} < b,\quad
  {n'' n'' \over m'' m''} > b.
   \eqno (240.)$$
For if the proposed ratio~$b$ be not the square of any whole or
fractional number, then the existence of such a ratio~$a$ may be
proved from the two preceding Lemmas, or from their Corollaries,
by observing that the relations (240.) give
$${n'' n'' \over m'' m''} > {n' n' \over m' m'},
   \quad\hbox{and therefore}\quad
  {n'' \over m''} > {n' \over m'};
   \eqno (241.)$$
so that no two conditions of the forms (239.) are incompatible
with each other, and there must be {\it at least one\/} positive
ratio~$a$ which satisfies them all.  And to prove in the same
case that there is {\it only one\/} such ratio, or that if any
one positive ratio~$a$ satisfy all the conditions (239.), no
greater ratio $c$ ($> a$) can possibly satisfy all those
conditions, we may observe that however little may be the excess
$\oppos a + c$ of the ratio~$c$ over $a$, this excess may be
multiplied by a positive whole number $m'$ so large that the
product shall be greater than unity, in such a manner that
$$m' (\oppos a + c) > 1,
   \eqno (242.)$$
and therefore
$$\oppos a + c > {1 \over m'},
   \quad\hbox{and}\quad
  c > {1 \over m'} + a;
   \eqno (243.)$$
and that then another positive (or null) whole number~$n'$ can be
so chosen that
$${n' n' \over m' m'} < b,\quad
  {1 + n' \over m'} \times {1 + n' \over m'} > b,
   \eqno (244.)$$
with which selection we shall have, by (239.) (240.) (243.),
$$a > {n' \over m'},\quad c > {1 + n' \over m'}:
   \eqno (245.)$$
whereas, if $c$ satisfied the conditions (239.) it ought to be
less than this fraction
$\displaystyle {1 + n' \over m'}$,
because the square of this positive fraction is greater by (244.)
than the proposed ratio~$b$.  In like manner it may be proved
that in the other case, when $b$ is the square of a positive
fractional or positive whole number
$\displaystyle {n \over m}$,
one positive ratio $a$ and only one, namely the number
$\displaystyle {n \over m}$
itself, will satisfy all the conditions (239.); in both cases,
therefore, the Lemma is true: and the consideration of the latter
case shows, that, under the conditions (239.),
$$a = {n \over m}
   \quad\hbox{if}\quad
  b = {nn \over mm},\quad {n \over m} > 0.
   \eqno (246.)$$
In no case do the conditions (239.) exclude {\it all\/}
ratios~$a$ whatever; but except in the case (246.) they
{\it exclude
all fractional ratios\/}: for it will soon be shown that the one
ratio~$a$ which they do not exclude has its square always $= b$,
and must, therefore, be an incommensurable number when $b$ is not
the square of any integer or fraction.  (Compare the
{\it Remark\/} annexed to the Corollary of the IInd Lemma.)

{\it Lemma\/}~IV.
If $b'$ and $b''$ be any two unequal positive ratios, it is
always possible to insert between them an intermediate fractional
ratio which shall be itself the square of another fractional
ratio
$\displaystyle {n \over m}$;
that is, we can always find, or conceive found, two positive
whole numbers $m$ and $n$ which shall satisfy the two conditions,
$${nn \over mm} > b',\quad
  {nn \over mm} < b'',
   \quad\hbox{if}\quad
   b'' > b',\quad
   b' > 0.
   \eqno (247.)$$
For, by the theorem of multiplication (200.), the square of the
fraction
$\displaystyle {1 + m' \over m}$
may be expressed as follows,
$${1 + n' \over m} \times {1 + n' \over m}
  = {1 \over mm} + {2n' \over mm} + {n' n' \over mm};
   \eqno (248.)$$
that is, its excess over the square of the fraction
$\displaystyle {n' \over m}$ is
$\displaystyle {1 \over mm} + {2 n' \over mm}$,
which is less than
$\displaystyle {2 \over m} \times {1 + n' \over m}$,
and constantly increases with the positive whole number~$n'$ when
the positive whole number~$m$ remains unaltered; so that the
$1 + n'$ squares of fractions with the common denominator~$m$, in
the following series,
$${1 \over m} \times {1 \over m},\quad
  {2 \over m} \times {2 \over m},\quad
  {3 \over m} \times {3 \over m},\quad\ldots\quad
  {n' \over m} \times {n' \over m},\quad
  {1 + n' \over m} \times {1 + n' \over m},
   \eqno (249.)$$
increase by increasing differences which are each less than
$\displaystyle {2 \over m} \times {1 + n' \over m}$,
and therefore than
$\displaystyle {1 \over k}$,
if we  choose $m$ and $n'$ so as to satisfy the conditions
$$m = 2ik,\quad 1 + n' = im,
   \eqno (250.)$$
$i$ and $k$ being any two positive whole numbers assumed at
pleasure: with this choice, therefore, of the numbers $m$ and
$m'$, some one (at least) such as $\displaystyle {nn \over mm}$
among the squares of fractions (249.), that is, some one among
the following squares of fractions,
$${1 \over 2ik} \times {1 \over 2ik},\quad
  {2 \over 2ik} \times {2 \over 2ik},\quad
  {3 \over 2ik} \times {3 \over 2ik},\quad\ldots\quad
  {2iik \over 2ik} \times {2iik \over 2ik},
   \eqno (251.)$$
of which the last is $=ii$, must lie between any two proposed
unequal positive ratios $b'$ and $b''$, of which the
greater~$b''$ does not exceed that last square~$ii$, and of which
the difference $\oppos b' + b''$ is not less than
$\displaystyle {1 \over k}$;
and positive whole numbers $i$ and $k$ can always be so chosen
as to satisfy these last conditions, however great the proposed
ratio~$b''$ may be, and however little may be its excess
$\oppos b' + b''$ over the other proposed ratio~$b'$.

\bigbreak

27.
With these preparations it is easy to prove, in a new and formal
way, the existence of {\it one determined positive square
root\/}~$\surd b$ for every proposed positive ratio~$b$, whether
that ratio~$b$ be or be not the square of any whole or of any
fractional number; for we can now prove this {\it Theorem\/}:

The square~$aa$ of the determined positive ratio~$a$, of which
ratio the existence was shown in the IIId.\ Lemma, is equal to
the proposed positive ratio~$b$ in the same Lemma; that is,
$$\left. \eqalign{
\hbox{if}\quad  a > {n' \over m'}
      \quad\hbox{whenever}\quad {n' n' \over m' m'} < b,\cr
\hbox{and}\quad a < {n'' \over m''}
      \quad\hbox{whenever}\quad {n'' n'' \over m'' m''} > b,\cr
\hbox{then}\quad aa = b,\quad a = \surd b,\cr}
   \right\}
   \eqno (252.)$$
$m'$~$n'$~$m''$~$n''$ being any positive whole numbers which
satisfy the conditions here mentioned, and $b$ being any
determined positive ratio.

For if the square~$aa$ of the positive ratio~$a$, determined by
these conditions, could be greater than the proposed positive
ratio~$b$, it would be possible, by the IVth Lemma, to insert
between them some positive fraction which would be the square of
another positive fraction
$\displaystyle {n \over m}$;
that is, we could choose $m$ and $n$ so that
$${nn \over mm} > b,\quad {nn \over mm} < aa:
   \eqno (253.)$$
and then, by the Corollary to the Ist Lemma, and by the
conditions (252.), we should be conducted to the two following
incompatible relations,
$${n \over m} < a,\quad a < {n \over m}.
   \eqno (254.)$$
A similar absurdity would result, if we were to suppose $aa$ less
than $b$; $aa$ must therefore be equal to $b$, that is, the
theorem is true.  It has, indeed, been here assumed as evident,
that every determined positive ratio~$a$ has a determined
positive square~$aa$; which is included in this more general but
equally evident principle, that any two determined positive
ratios or numbers have a determined positive product.

We find it, therefore, proved, by the most minute and rigorous
examination, that if we conceive any positive ratio~$x$ or $a$ to
increase constantly and continuously from $0$, we must conceive
its square $xx$ or $aa$ to increase constantly and continuously
with it, so as to pass successively but only once through every
state of positive ratio~$b$: and therefore that every determined
positive ratio~$b$ has one determined positive square root
$\surd b$, which will be commensurable or incommensurable,
according as $b$ can or cannot be expressed as the square of a
fraction.  When $b$ cannot be so expressed, it is still possible
to {\it approximate in fractions\/} to the incommensurable square
root~$\surd b$, by choosing successively large and larger
positive denominators, and then seeking for every such
denominator~$m'$ the corresponding positive numerator~$n'$ which
satisfies the two conditions (244.); for although every fraction
thus found will be less than the sought root $\surd b$, yet the
error, or the positive correction which must be added to it in
order to produce the accurate root $\surd b$, is less than the
reciprocal of the denominator~$m'$, and may be made as little
different as we please from $0$, (though it can never be made
exactly $=0$), by choosing that denominator large enough.  This
process of approximation to an incommensurable root~$\surd b$ is
capable, therefore, of an indefinitely great, though never of a
perfect accuracy; and using the notation already given for
{\it limits}, we may write
$$\surd b = \underline{\rm L} {n' \over m'},
   \quad\hbox{if}\quad
   {n' n' \over m' m'} < b,\quad
   {1 + n' \over m'} \times {1 + n' \over m'} > b,
   \eqno (255.)$$
and may think of the incommensurable root as the {\it limit\/} of
the varying fractional number.

The only additional remark which need be made, at present, on the
subject of the progression of the square~$xx$, or $aa$, as
depending on the progression of the root~$x$, or $a$, is that
since (by the 24th article) the square remains positive and
unchanged when the root is changed from positive to
contra-positive, in such a manner that
$$\oppos a \times \oppos a = a \times a,
   \eqno (256.)$$
the square~$aa$ must be conceived as {\it first\/} constantly and
continuously {\it decreasing\/} or {\it retrograding\/} towards
$0$, and {\it afterwards\/} constantly and continuously
{\it increasing\/} or {\it advancing\/} from $0$, if the root~$a$
be conceived as constantly and continuously increasing or
advancing, in the general progression of ratio, from states
indefinitely far from $0$ on the contra-positive side, to other
states indefinitely far from $0$, but on the positive side in the
progression.

\bigbreak

\centerline{\it
On Continued Analogies, or Series of Proportional Steps; and on
Powers, and Roots, and}

\nobreak\vskip 3pt

\centerline{\it
Logarithms of Ratios.}

\nobreak\bigskip

28.
Four effective steps $a$~$b$~$b'$~$b''$ may be said to form a
continued analogy or continued proportion, $a$ and $b''$ being
the extremes, and $b$ and $b'$ the means, when they are connected
by one common ratio in the following manner:
$${{\sr b}'' \over {\sr b}'} = {{\sr b}' \over {\sr b}}
   = {{\sr b} \over {\sr a}};
   \eqno (257.)$$
and if we denote for abridgement this common ratio by $a$, we may
write
$${\sr b}   = a \times {\sr a},\quad
  {\sr b}'  = a \times a \times {\sr a},\quad
  {\sr b}'' = a \times a \times a \times {\sr a}.
   \eqno (258.)$$
Reciprocally, when $b$~$b'$~$b''$ can be thus expressed, the four
steps ${\sr a}$~${\sr b}$~${\sr b}'$~${\sr b}''$ compose a
continued {\it analogy\/}; and it is clear that if the first
extreme step~${\sr a}$ and the common ratio~$a$ be given, the
other steps can be deduced by the multiplications (258.).  It is
easy also to perceive, that if the two extremes ${\sr a}$ and
${\sr b}''$ be given, the two means ${\sr b}$ and ${\sr b}'$ may
be conceived to be determined (as necessarily connected with
these) in one and in only one way; and thus that the insertion of
{\it two mean proportionals\/} between two given effective steps,
is never impossible nor ambiguous, like the insertion of a single
mean proportional.  In fact, it follows from the theorems of
multiplication that the product $a \times a \times a$, which may
be called the {\it cube\/} of the number or ratio~$a$, is not
obliged (like the square $a \times a$) to be always a positive
ratio, but is positive or contra-positive according as $a$ itself
(which may be called the {\it cube-root\/} of this product
$a \times a \times a$) is positive or contra-positive; and on
examining the law of its progression, (as we lately examined the
law of the progression of the square,) we find that the cube
$a \times a \times a$ increases constantly and continuously with
its cube-root~$a$ from states indefinitely far from zero, on the
contra-positive side, to states indefinitely far advanced on the
positive side of zero, in the general progression of ratio, so as
to pass successively but only once through every state of
contra-positive or positive ratio, instead of first decreasing or
retrograding, and afterwards increasing or advancing, like
the square.  Thus every ratio has one and only one cube-root,
(commensurable or incommensurable,) although a ratio has
sometimes two square-roots and sometimes none, according as it is
positive or contra-positive; and when the two extreme effective
steps ${\sr a}$ and ${\sr b}''$ of the continued analogy (257.)
are given, we can always conceive the cube-root~$a$ of their
ratio $\displaystyle {{\sr b}'' \over {\sr a}}$ determined, and
hence the two mean steps or mean proportionals of the analogy,
${\sr b}$ and ${\sr b}'$.

\bigbreak

29.
In general, as we conceived a continued analogy or {\it series of
equi-distant moments}, generated from a single standard
moment~${\sc a}$, by the {\it repetition\/} of a forward
{\it step\/}~${\sr a}$ and of a backward step~$\oppos {\sr a}$;
so we may now conceive, as another sort of continued analogy, a
{\it series of proportional steps}, generated from a single
standard (effective) step~${\sr a}$, by the {\it repetition\/}
of the {\it act of multiplication\/} which corresponds to and is
determined by some one multiplier or ratio $a$ ($\, \neq 0$), and
of the inverse or reciprocal act of multiplication determined by
the reciprocal multiplier or ratio $\recip a$: namely the
following series of proportional steps,
$$\ldots \, \recip a \times \recip a \times \recip a \times {\sr a},\quad
  \recip a \times \recip a \times {\sr a},\quad
  \recip a \times {\sr a},\quad
  {\sr a},\quad
  a \times {\sr a},\quad
  a \times a \times {\sr a},\quad
  a \times a \times a \times {\sr a},\ldots
   \eqno (259.)$$
which may also be thus denoted,
$$\ldots \, \recip (aaa) \times {\sr a},\quad
  \recip (aa) \times {\sr a},\quad
  \recip a \times {\sr a},\quad
  1 \times {\sr a},\quad
  a \times {\sr a},\quad
  aa \times {\sr a},\quad
  aaa \times {\sr a},\ldots
   \eqno (260.)$$
and in which we may consider the system or series of ratios or
multipliers,
$$\ldots \, \recip (aaa),\quad
  \recip (aa),\quad
  \recip a,\quad
  1,\quad
  a,\quad
  aa,\quad
  aaa,\ldots
   \eqno (261.)$$
to be a {\it system generated\/} from the original ratio or
multiplier~$a$, by a {\it system of acts\/} of generation having
all one common character: as we before considered the system of
multiple steps (98.),
$$\ldots \,
   \oppos {\sr a} + \oppos {\sr a} + \oppos {\sr a},\quad
   \oppos {\sr a} + \oppos {\sr a},\quad
   \oppos {\sr a},\quad
   0,\quad
   {\sr a},\quad
   {\sr a} + {\sr a},\quad
   {\sr a} + {\sr a} + {\sr a},\ldots$$
to be a system of steps generated from the original
step~${\sr a}$ by a system of acts of generation to which we gave
the common name of acts of multiplying.

In conformity with this conception, we may call the original
ratio~$a$ the {\it base\/} of the system of ratios (261.) and may
call those ratios by the common name of {\it powers\/} of that
common base, and say that they are (or may be) formed by acts of
{\it powering\/} it.  And to distinguish any one such power, or
one such act of powering, from all the other powers in the
system, or from all the other acts of powering, we may employ the
aid of {\it determining numbers\/}, ordinal or cardinal, in a
manner analogous to that explained in the 13th article for a
system of multiple steps.  Thus, we may call the ratios
$a, aa, aaa,\ldots$ by the common name of {\it positive powers\/}
of the base~$a$, and may distinguish them by the special ordinal
names {\it first}, {\it second}, {\it third}, \&c.; so that the
ratio~$a$ is, in this view, its own first positive power; the
second positive power is the square~$aa$, and the third positive
power is the cube.  Again, we may call the ratio~$1$, which
immediately precedes these positive powers in the series, the
{\it zero-power\/} of the base~$a$, by analogy to the
zero-multiple in the series of multiple steps, which immediately
preceded in that series the system of positive multiples: and the
ratios $\recip a, \recip (aa), \recip (aaa),\ldots$
which precede this zero-power~$1$ in the series of powers (261.),
may be called, by the same analogy, from their order of position,
{\it contra-positive powers\/} of $a$, so that the reciprocal
$\recip a$ of any ratio~$a$ is the {\it first contra-positive
power\/} of that ratio, the reciprocal $\recip (aa)$ of its
square is its second contra-positive power, and so on.  We may
also distinguish the several corresponding acts of powering by
the corresponding cardinal numbers, positive, or contra-positive,
or null, and may say (for example) that the third positive power
$aaa$ is formed from the base~$a$ by the act of {\it powering by
positive three\/}; that the second contra-positive power
$\recip (aa)$ is formed from the same base~$a$ by {\it powering
by contra-positive two\/}; and that the zero-power~$1$ is (or may
be) formed from $a$ by powering that base by the cardinal number
or number {\it none}.  In written symbols, answering to these
thoughts and names, we may denote the {\it series of powers\/}
(261.), and the {\it series of proportional steps\/} (260.), as
follows,
$$\ldots \, a^{\oppos 3}, a^{\oppos 2}, a^{\oppos 1},
   a^0, a^1, a^2, a^3,\ldots
   \eqno (262.)$$
and
$$\ldots \, a^{\oppos 3} \times {\sr a},\quad
   a^{\oppos 2} \times {\sr a},\quad
   a^{\oppos 1} \times {\sr a},\quad
   a^0 \times {\sr a},\quad
   a^1 \times {\sr a},\quad
   a^2 \times {\sr a},\quad
   a^3 \times {\sr a},\ldots
   \eqno (263.)$$
in which
$$a^0 = 1,
   \eqno (264.)$$
and
$$\left. \multieqalign{
a^1 &= a, &   a^{\oppos 1} &= \recip a,\cr
a^2 &= aa, &  a^{\oppos 2} &= \recip (aa),\cr
a^3 &= aaa, & a^{\oppos 3} &= \recip (aaa),\cr
    &\hbox{\&c.} &         &\hbox{\&c.}\cr}
   \right\}
   \eqno (265.)$$
And we may give the name of {\it exponents\/} or
{\it logarithms\/} to the determining numbers, ordinal or
cardinal,
$$\ldots \, \oppos 3,\quad
   \oppos 2,\quad
   \oppos 1,\quad
   0,\quad
   1,\quad
   2,\quad
   3,\ldots
   \eqno (266.)$$
which answer the question ``{\it which in order\/} is the
Power?'' or this other question ``{\it Have any\/} (effective)
acts of multiplication, equivalent or reciprocal to the original
act of multiplying by the given ratio~$a$, been combined to
produce the act of multiplying by the Power; and if any, then
{\it How many}, and {\it In which direction}, that is, whether
they are similar or opposite in effect, (as enlarging or
diminishing the step on which they are performed,) to that
original act?''  Thus $2$ is the logarithm of the square or
second power~$aa$, when compared with the base~$a$; $3$ is the
logarithm of the cube~$aaa$, $1$ is the logarithm of the base~$a$
itself, $\oppos 1$ is the logarithm of the reciprocal~$\recip a$,
and $0$ is the logarithm of the ratio~$1$ considered as the
zero-power of $a$.

With these conceptions and notations of powers and logarithms, we
can easily prove the relation
$$a^\nu \times a^\mu = a^{\nu + \mu},
   \eqno (267.)$$
for any integer logarithms $\mu$ and $\nu$, whether positive, or
contra-positive, or null; and this other connecting relation
$$b^\nu = a^{\nu \times \mu} \quad\hbox{if}\quad b = a^\mu;
   \eqno (268.)$$
which may be thus expressed in words: ``Any two powers of any
common base may be multiplied together by adding their
logarithms,'' and ``Any proposed power may be powered by any
proposed whole number, by multiplying its logarithm by that
number,'' if the sum of the two proposed logarithms in the first
case, or the multiple of the proposed logarithm in the second
case, be employed as a new logarithm, to form a new power of the
original base or ratio; the logarithms here considered being all
whole numbers.

\bigbreak

30.
The act of passing from a base to a power, is connected with an
inverse or reciprocal act of returning from the power to the
base; and the conceptions of both these acts are included in the
more comprehensive conception of the act of passing from any one
to any other of the ratios of the series (261.) or (262.).  This
act of passing from any one power~$a^\mu$ to any other
power~$a^{\nu}$ of a common base~$a$, may be still called in
general an act of {\it powering\/}; and more particularly,
(keeping up the analogy to the language already employed in the
theory of multiple steps,) it may be called the act of
{\it powering by the fractional
number\/}~$\displaystyle {\nu \over \mu}$.  By the same analogy
of definition, this fractional number may be called the
{\it logarithm\/} of the resulting power, and the power itself
may be denoted in written symbols as follows,
$$(a^\mu)^{\nu \over \mu} = a^\nu,
   \eqno (269.)$$
or thus,
$$c = b^{\nu \over \mu},
   \quad\hbox{if}\quad b = a^\mu,\quad c = a^\nu.
   \eqno (270.)$$

In the particular case when the numerator~$\nu$ is~$1$, and when,
therefore, we have to power by the reciprocal of a whole number,
we may call the result
$\displaystyle (a^\mu)^{1 \over \mu}$,
that is $a^1$, $= a$, a {\it root\/} or more fully {\it the
$\mu$'th root\/} of the power or ratio~$a^\mu$; and we may call
the corresponding act of powering, an {\it extraction of the
$\mu$'th root}, or a {\it rooting by the (whole) number\/}~$\mu$.
Thus, to power any proposed ratio~$b$ by the reciprocal number
$\displaystyle {1 \over 2}$ or $\displaystyle {1 \over 3}$,
is to extract the second or the third root, that is, (by what has
been already shown,) the square-root or the cube-root, of $b$, or
to root the proposed ratio~$b$ by the number $2$ or $3$; and in
conformity with this last mode of expression, the following
notation may be employed,
$$a = \root \mu \of{\vphantom{b}} b
   \quad\hbox{when}\quad
  b = a^\mu,\quad a = b^{1 \over \mu}:
   \eqno (271.)$$
so that a square-root $\surd b$ may also be denoted by the symbol
$\root 2 \of{\vphantom{b}} b$, and the cube-root of $b$ may be
denoted by $\root 3 \of{\vphantom{b}} b$.  And whereas we saw, in
considering square-roots that a contra-positive ratio $b < 0$ has
no square-root, and that a positive ratio $b > 0$ has two
square-roots, one positive $= \surd b$ and the other
contra-positive $= \oppos \surd b$, of which each has its square
$= b$; we may consider the new sign $b^{1 \over 2}$ or
$\root 2 \of{\vphantom{b}} b$ as denoting indifferently either of
these two roots, reserving the old sign $\surd b$ to denote
specially that one of them which is positive, and the other old
sign $\oppos \surd b$ to denote specially that one of them which
is contra-positive.  Thus $\surd b$ and $\oppos \surd b$ shall
still remain determinate signs, implying each a determinate
ratio, (when $b > 0$,) while $\root 2 \of{\vphantom{b}} b$ and
$b^{1 \over 2}$ shall be used as ambiguous signs, susceptible
each of two different meanings.  But $\root 3 \of{\vphantom{b}}
b$ is a determinate sign, because a ratio has only one cube-root.
In general, an {\it even\/} root, such as the second, fourth, or
sixth, of a proposed ratio~$b$, is ambiguous if that ratio be
positive, and impossible if $b$ be contra-positive; because an
even power, or a power with an even integer for its logarithm, is
always a positive ratio, whether the base be positive or
contra-positive: but an {\it odd\/} root, such as the third or
fifth, is always possible and determinate.

\bigbreak

31.
It may, however, be useful to show more distinctly, by a method
analogous to that of the 26th and 27th articles, that for any
proposed positive ratio~$b$ whatever, and for any positive whole
number~$m$, it is possible to determine, or conceive determined,
one positive ratio~$a$, and only one, which shall have its $m$'th
power $= b$; and for this purpose to show that the power $a^m$
increases constantly and continuously from zero with $a$, so as
to pass successively, but only once, through every state of
positive ratio~$b$.  On examining the proof already given of this
property, in the particular case of the power~$a^2$, we see that
in order to extend that proof to the more general case of the
power~$a^m$, we have only to generalise, as follows, the Ist,
IIId and IVth Lemmas, and the Corollary, of the Ist, with the
Theorem resulting from all four, retaining the IInd Lemma.

Vth {\it Lemma\/}: (generalised from Ist.)
$$\hbox{If}\quad
   y \gteqlt x,
   \quad\hbox{and}\quad
   x > 0,\quad y > 0,
   \quad\hbox{then}\quad
   y^m \gteqlt x^m.
   \eqno (272.)$$
When $m = 1$, this Lemma is evident, because the first powers
$y^1$ and $x^1$ coincide with the ratios $y$ and $x$.  When
$m > 1$, the Lemma may be easily deduced from the conceptions of
ratios, and of powers with positive integer exponents; it may
also be proved by observing that the difference
$\oppos x^m + y^m$, between the powers $x^m$ and $y^m$, in which
the symbol $\oppos x^m$ denotes the same thing as if we had
written more fully $\oppos (x^m)$, and which may be obtained in
one way by the subtraction of $x^m$ from $y^m$ may also be
obtained in another way by multiplication from the difference
$\oppos x + y$ as follows:
$$\oppos x^m + y^m
   = (\oppos x + y) \times (
            x^{\oppos 1 + m} y^0
          + x^{\oppos 2 + m} y^1
          + \ldots
          + x^1 y^{\oppos 2 + m}
          + x^0 y^{\oppos 1 + m} ),
   \eqno (273.)$$
and is, therefore, positive, or contra-positive, or null,
according as the difference $\oppos x + y$ of the positive ratios
$x$ and $y$ themselves is positive, or contra-positive, or null,
because the other factor of the product (273.) is positive.  For
example,
$$\oppos x^3 + y^3 = ( \oppos x + y ) \times (x^2 + xy + y^2);
   \eqno (274.)$$
and, therefore, when $x$ and $y$ and consequently
$x^2 + xy + y^2$ are positive, the difference
$\oppos x^3 + y^3$ and the difference $\oppos x + y$ are
positive, or contra-positive, or null together.

As a {\it Corollary\/} of this Lemma, we see that, conversely,
$$\hbox{if}\quad
   y^m \gteqlt x^m,
   \quad\hbox{and}\quad
   x > 0,\quad y > 0,
   \quad\hbox{then}\quad
   y \gteqlt x.
   \eqno (275.)$$

Thus the power $x^m$ an the root~$x$ increase {\it constantly\/}
together, when both are positive ratios.

The {\it logic\/} of this last deduction, of the Corollary (275.)
from the Lemma (272.), must not be confounded with that erroneous
form of argument which infers the truth of the antecedent of a
true hypothetical proposition from the truth of the consequent;
that is, with the too common {\it sophism\/}: If ${\sc a}$ be
true then ${\sc b}$ is true; but ${\sc b}$ is true, therefore
${\sc a}$ is true.  The Lemma (272.) asserts three hypothetical
propositions, which are tacitly supposed to be each transformed,
or logically converted, according to this {\it valid\/}
principle, that the falsehood of the consequent of a true
hypothetical proposition infers the falsehood of the antecedent;
or according to this just formula: If ${\sc a}$ were true then
${\sc b}$ would be true; but ${\sc b}$ is false, therefore
${\sc a}$ is not true.  Applying this just principle to each of
the three hypothetical propositions of the Lemma, we are entitled
to infer, by the general principles of Logic, these three
converse hypothetical propositions:
$$\left. \eqalign{
\hbox{if}\quad y^m \ngt x^m
   \quad\hbox{then}\quad y \ngt x;\cr
\hbox{if}\quad y^m \neq x^m
   \quad\hbox{then}\quad y \neq x;\cr
\hbox{if}\quad y^m \nlt x^m
   \quad\hbox{then}\quad y \nlt x;\cr}
   \right\}
   \eqno (276.)$$
$x$ and $y$ being here any positive ratios, and $m$ any positive
whole number, and the signs $\ngt$ and $\nlt$ denoting
respectively ``not~$>$'' and ``not~$<$'' as the sign $\neq$
denotes ``not~$=$''.  And if, to the propositions (276.), we join
this principle of intuition in Algebra, as the Science of Pure
Time, that a variable moment~${\sc b}$ must either follow, or
coincide with, or precede a given or variable moment~${\sc a}$,
but cannot do two of these three things at once, and therefore
(by the 21st article) that a variable ratio~$y$ must also bear
one but only one of these three ordinal relations to a given or
variable ratio~$x$, which shows that
$$\left. \eqalign{
\hbox{when}\quad y^m > x^m
   \quad\hbox{then}\quad y^m \neq x^m
   \quad\hbox{and}\quad  y^m \nlt x^m,\cr
\hbox{when}\quad y^m = x^m
   \quad\hbox{then}\quad y^m \nlt x^m
   \quad\hbox{and}\quad  y^m \ngt x^m,\cr
\hbox{when}\quad y^m < x^m
   \quad\hbox{then}\quad y^m \ngt x^m
   \quad\hbox{and}\quad  y^m \neq x^m,\cr}
   \right\}
   \eqno (277.)$$
and that
$$\left. \eqalign{
\hbox{when}\quad y \neq x
   \quad\hbox{and}\quad  y \nlt x
   \quad\hbox{then}\quad y > x,\cr
\hbox{when}\quad y \nlt x
   \quad\hbox{and}\quad  y \ngt x
   \quad\hbox{then}\quad y = x,\cr
\hbox{when}\quad y \ngt x
   \quad\hbox{and}\quad  y \neq x
   \quad\hbox{then}\quad y < x,\cr}
   \right\}
   \eqno (278.)$$
we find finally that the Corollary (275.) is true.  The same
logic was tacitly employed in deducing the Corollary of the Ist
Lemma, in the hope that it would be mentally supplied by the
attentive reader.  It has now been stated expressly, lest any
should confound it with that dangerous and common fallacy, of
inferring, in Pure Science, the necessary truth of a premiss in
an argument, from the known truth of the conclusion.

Resuming the more mathematical part of the research, we may next
establish this

\smallbreak

VIth {\it Lemma\/} (generalised from IIId):
There exists one positive ratio~$a$, and only one, which
satisfies all the following conditions,
$$\left. \eqalign{
a > {n' \over m'}
   \quad\hbox{whenever}\quad
   \left( {n' \over m'} \right)^m < b,\cr
a < {n'' \over m''}
   \quad\hbox{whenever}\quad
   \left( {n'' \over m''} \right)^m > b;\cr}
   \right\}
   \eqno (279.)$$
$b$ being any given positive ratio, and $m$ any given positive
whole number, while $m' \, n' \, m'' \, n''$ are also positive
but variable whole numbers.  The proof of this Lemma is so like
that of the IIId, that it need not be written here; and it shows
that in the particular case when the given ratio~$b$ is the
$m^{\rm th}$ power of a positive fraction
$\displaystyle {n_\prime \over m_\prime}$,
then $a$ is that fraction itself.  In general, it will soon be
shown that under the conditions of this Lemma the $m^{\rm th}$
power of $a$ is $b$.

VIIth {\it Lemma\/} (generalised from IVth).
It is always possible to find, or to conceive as found, two
positive whole numbers $m_\prime$ and $n_\prime$, which shall
satisfy the two conditions
$$\left( {n_\prime \over m_\prime} \right)^m > b',\quad
  \left( {n_\prime \over m_\prime} \right)^m < b'',
   \quad\hbox{if}\quad b'' > b',\quad b' > 0,
   \eqno (280.)$$
$m$ being any given positive whole number; that is, we can insert
between any two unequal positive ratios $b'$ and $b''$ an
intermediate fractional ratio which is itself the $m^{\rm th}$
power of a fraction.

For, when $m = 1$, this Lemma reduces itself to the IInd; and
when $m > 1$, the theorem (273.) shows that the excess of
$\displaystyle \left( {1 + n \over m_\prime} \right)^n$ over
$\displaystyle \left( {n \over m_\prime} \right)^n$
may be expressed as follows:
$$\oppos \left( {n \over m_\prime} \right)^m
       + \left( {1 + n \over m_\prime} \right)^m
   = {1 \over m_\prime'} \times p,
   \eqno (281.)$$
in which
$$\eqalignno{
p  &=    \left( {n \over m_\prime} \right)^{\oppos 1 + m}
       + \left( {n \over m_\prime} \right)^{\oppos 2 + m}
            \left( {1 + n \over m_\prime} \right)
       + \left( {n \over m_\prime} \right)^{\oppos 3 + m}
            \left( {1 + n \over m_\prime} \right)^2 \cr
   &\mathrel{\phantom{=}} \mathord{}
       + \ldots
       + \left( {n \over m_\prime} \right)^2
            \left( {1 + n \over m_\prime} \right)^{\oppos 3 + m}
       + {n \over m_\prime}
            \left( {1 + n \over m_\prime} \right)^{\oppos 2 + m}
       + \left( {1 + n \over m_\prime} \right)^{\oppos 1 + m};
   &(282.)\cr}$$
for example, when $m = 3$, the excess of the cube
$\displaystyle \left( {1 + n \over m_\prime} \right)^3$
over the cube
$\displaystyle \left( {n \over m_\prime} \right)^3$, is
$$\oppos \left( {n \over m_\prime} \right)^3
       + \left( {1 + n \over m_\prime} \right)^3
   = {1 \over m_\prime} \times
         \left\{
            \left( {n \over m_\prime} \right)^2
          + {n \over m_\prime} {1 + n \over m_\prime}
          + \left( {1 + n \over m_\prime} \right)^2
         \right\}.
   \eqno (283.)$$

In general, the number of the {\it terms\/} (or added parts) in
the expression (282.), is $m$, and they are all unequal, the
least being
$\displaystyle \left( {n \over m_\prime} \right)^{\oppos 1 + m}$,
and the greatest being
$\displaystyle \left( {1 + n \over m_\prime} \right)^{\oppos 1 + m}$;
their sum, therefore, is less than the $m^{\rm th}$ multiple of
this greatest term, that is,
$$p < m \times \left( {1 + n \over m_\prime} \right)^{\oppos 1 + m},
   \eqno (284.)$$
and therefore the excess (281.) is subject to the corresponding
condition
$$\oppos \left( {n \over m_\prime} \right)^m
       + \left( {1 + n \over m_\prime} \right)^m
   <  {m \over m_\prime} 
         \left( {1 + n \over m_\prime} \right)^{\oppos 1 + m};
   \eqno (285.)$$
for example,
$$\oppos \left( {n \over m_\prime} \right)^3
       + \left( {1 + n \over m_\prime} \right)^3
   <  {3 \over m_\prime} 
         \left( {1 + n \over m_\prime} \right)^2.
   \eqno (286.)$$
However, this excess (281.) increases constantly with $n$, when
$m_\prime$ remains unaltered, because $p$ so increases; so that
the $1 + n$ fractions of the series
$$    \left( {1 \over m_\prime} \right)^m,\quad
      \left( {2 \over m_\prime} \right)^m,\quad
      \left( {3 \over m_\prime} \right)^m,\quad\ldots\quad
      \left( {1 + n \over m_\prime} \right)^m,
   \eqno (287.)$$
increase by increasing differences, (or advance by increasing
intervals,) which are each less than
$\displaystyle {m \over m_\prime}
      \left( {1 + n \over m_\prime} \right)^{\oppos 1 + n}$,
and therefore than
$\displaystyle {1 \over k}$,
if we choose $m_\prime$ and $n$ to satisfy the conditions
$$1 + n = i m_\prime,\quad
  m_\prime = k m \times i^{\oppos 1 + m}
   =  {k m i^m \over i},
   \eqno (288.)$$
$i$ and $k$ being any two positive whole numbers assumed at
pleasure; with this choice, therefore, of the numbers $m_\prime$
and $n$, some one (at least), such as
$\displaystyle \left( {n_\prime \over m_\prime} \right)^m$,
of the series of powers of fractions (287.), of which the last is
$= i^m$, will fall between any two proposed unequal positive
ratios $b'$ and $b''$, if the greater~$b''$ does not exceed that
last power $i^m$, and if the difference
$\oppos b' + b''$ is not less than
$\displaystyle {1 \over k}$;
and these conditions can be always satisfied by a suitable choice
of the whole numbers $i$ and $k$, however large may be the given
greater positive ratio~$b''$, and however little may be its given
excess over the lesser positive ratio~$b'$.

Hence, finally, this Theorem:
$$\left. \eqalign{
\hbox{If}\quad
      a > {n' \over m'},
   \quad\hbox{and}\quad
      a < {n'' \over m''},\cr
   \quad\hbox{whenever}\quad
      \left( {n' \over m'} \right)^m < b,\quad
      \left( {n'' \over m''} \right)^m > b,\cr
   \quad\hbox{then}\quad
      a^m = b,\quad
      a = \root m \of{\vphantom{b}} b = b^{1 \over m};\cr}
   \right\}
   \eqno (289.)$$
$b$ denoting any given positive ratio, and $m$ any given positive
whole number, while $m'$~$n'$~$m''$~$n''$ are any arbitrary
positive whole numbers which satisfy these conditions, and $a$ is
another positive ratio which the VIth Lemma shows to be
determined.

For if $a^m$ could be $> b$, we could, by the VIth Lemma, insert
between them a positive fraction of the form
$\displaystyle \left( {n_\prime \over m_\prime} \right)^m$,
such that
$$\left( {n_\prime \over m_\prime} \right)^m > b,\quad
  \left( {n_\prime \over m_\prime} \right)^m < a^m;
   \eqno (290.)$$
and then by the Corollary of the Vth Lemma, and by the conditions
(289.), we should deduce the two incompatible relations
$${n_\prime \over m_\prime} < a,\quad
  a < {n_\prime \over m_\prime},
   \eqno (291.)$$
which would be absurd.  A similar absurdity would follow from
supposing that $a^m$ could be less than $b$; $a^m$ must therefore
be $= b$, that is, the Theorem is true.  It has, indeed, been all
along assumed as evident that every determined positive ratio~$a$
has a determined positive $m^{\rm th}$ power~$a^m$, when $m$ is a
positive whole number; which is included in this more general but
also evident principle, that any $m$ determined positive ratios
or numbers have a determined positive product.

Every positive ratio~$b$ has therefore one, and only one,
positive ratio~$a$ for its $m^{\rm th}$ root, which is
commensurable or incommensurable, according as $b$ can or cannot
be put under the form
$\displaystyle \left( {n_\prime \over m_\prime} \right)^m$;
but which, when incommensurable, may be theoretically conceived
as the accurate limit of a variable fraction,
$$a = \root m \of{\vphantom{b}} b
    = \underline{\rm L} {n' \over m'},
   \quad\hbox{if}\quad
      \left( {n' \over m'} \right)^m < b,\quad
      \left( {1 + n' \over m'} \right)^m > b,
   \eqno (292.)$$
and may be practically approached to, by determining such
fractions
$\displaystyle {n' \over m'}$,
with larger and larger whole numbers $m'$ and $n'$ for their
denominators and numerators.  And whether $m$ be odd or even, we
see that the power~$a^m$ increases {\it continuously\/} (as well
as constantly) with its positive root or base~$a$, from zero up
to states indefinitely greater and greater.  But if this root, or
base, or ratio~$a$ be conceived to advance constantly and
continuously from states indefinitely far from zero on the
contra-positive side to states indefinitely far upon the positive
side, then the power~$a^m$ will either advance constantly and
continuously likewise, though not with the same quickness, from
contra-positive to positive states, or else will first constantly
and continuously retrograde to zero, and afterwards advance from
zero, remaining always positive, according as the positive
exponent or logarithm~$m$ is an odd or an even integer.  It is
understood that for any such positive exponent~$m$,
$$0^m = 0,
   \eqno (293.)$$
the powers of $0$ with positive integer exponents being
considered as all themselves equal to $0$, because the repeated
multiplication by this null ratio generates from any one
effective step~${\sr a}$ the series of proportional steps,
$${\sr a},\quad
   0 \times {\sr a} = 0,\quad
   0 \times 0 \times {\sr a} = 0,\quad\ldots,
   \eqno (294.)$$
which may be continued indefinitely {\it in one direction\/}, and
in which all steps after the first are null; although we were
obliged to exclude the consideration of such null ratios in
forming the series (259.) because we wished to continue that
series of steps indefinitely in two opposite directions.

\bigbreak

32.
We are now prepared to discuss completely the meaning, or
meanings, if any, which ought to be assigned to any proposed
symbol of the class $b^{\nu \over \mu}$, $b$ denoting any
proposed ratio, and $\mu$ and $\nu$ any proposed whole numbers.
By the 30th article, the symbol $b^{\nu \over \mu}$ denotes
generally the $\nu$'th power of a ratio~$a$ of which $b$ is the
$\mu$'th power, or, in other words, the $\nu$th power of a
$\mu$th root of $b$; so that the mental operation of passing from
the ratio~$b$ to the ratio $b^{\nu \over \mu}$, is compounded,
(which it can be performed at all,) of the two operations of
first rooting by the one whole number~$\mu$, and then powering by
the other whole number~$\nu$: and we may write,
$$b^{\nu \over \mu}
   = ( \root \mu \of{\vphantom{b}} b )^\nu
   = ( b^{1 \over \mu} )^\nu.
   \eqno (295.)$$
The ratio~$b$, and the whole numbers $\mu$ and $\nu$, may each be
either positive, or contra-positive, or null; and thus there
arise many cases, which may be still farther subdivided, by
distinguishing between odd and even values of the positive or
contra-positive whole numbers.  For, if we suppose that ${\rm B}$
denotes a positive ratio, and that $m$ and $n$ denote positive
whole numbers, we may then suppose
$$\left. \eqalign{
b = {\rm B},
   \quad\hbox{or}\quad b = 0,
   \quad\hbox{or}\quad b = \oppos {\rm B},\cr
\mu = m,
   \quad\hbox{or}\quad \mu = 0,
   \quad\hbox{or}\quad \mu = \oppos m,\cr
\nu = n,
   \quad\hbox{or}\quad \nu = 0,
   \quad\hbox{or}\quad \nu = \oppos n,\cr}
   \right\}
   \eqno (296.)$$
and thus shall obtain the twenty-seven cases following,
$$\left. \eqalign{
{\rm B}^{n \over m},\quad
      {\rm B}^{0 \over m},\quad
      {\rm B}^{\oppos n \over m},\cr
{\rm B}^{n \over 0},\quad
      {\rm B}^{0 \over 0},\quad
      {\rm B}^{\oppos n \over 0},\cr
{\rm B}^{n \over \oppos m},\quad
      {\rm B}^{0 \over \oppos m},\quad
      {\rm B}^{\oppos n \over \oppos m},\cr}
   \right\}
   \eqno (297.)$$
$$\left. \eqalign{
0^{n \over m},\quad
      0^{0 \over m},\quad
      0^{\oppos n \over m},\cr
0^{n \over 0},\quad
      0^{0 \over 0},\quad
      0^{\oppos n \over 0},\cr
0^{n \over \oppos m},\quad
      0^{0 \over \oppos m},\quad
      0^{\oppos n \over \oppos m},\cr}
   \right\}
   \eqno (298.)$$
$$\left. \eqalign{
(\oppos {\rm B})^{n \over m},\quad
      (\oppos {\rm B})^{0 \over m},\quad
      (\oppos {\rm B})^{\oppos n \over m},\cr
(\oppos {\rm B})^{n \over 0},\quad
      (\oppos {\rm B})^{0 \over 0},\quad
      (\oppos {\rm B})^{\oppos n \over 0},\cr
(\oppos {\rm B})^{n \over \oppos m},\quad
      (\oppos {\rm B})^{0 \over \oppos m},\quad
      (\oppos {\rm B})^{\oppos n \over \oppos m},\cr}
   \right\}
   \eqno (299.)$$
which we may further sub-divide by putting $m$ and $n$ under the
forms
$$\left. \eqalign{
m = 2i,\quad\hbox{or}\quad m = \oppos 1 + 2i,\cr
n = 2k,\quad\hbox{or}\quad n = \oppos 1 + 2k,\cr}
   \right\}
   \eqno (300.)$$
in which $i$ and $k$ themselves denote positive whole numbers.
But, various as these cases are, the only difficulty in
discussing them arises from the occurrence, in some, of the ratio
or number~$0$; and to remove this difficulty, we may lay down the
following rules, deduced from the foregoing principles.

To power the ratio~$0$ by any positive whole number~$m$, gives,
by (293.), the ratio~$0$ as the result.  This ratio~$0$ is,
therefore, at least {\it one\/} $m$'th root of $0$; and since no
positive or contra-positive ratio can thus give $0$ when powered
by any positive whole number, we see that the {\it only\/} $m$'th
root of $0$ is $0$ itself.  Thus,
$$0^{1 \over m} = 0,
   \eqno (301.)$$
and generally,
$$0^{n \over m} = 0.
   \eqno (302.)$$

To power any positive ratio~$a$, whether positive, or
contra-positive, or null, by the number or logarithm~$0$, may be
considered to give $1$ as the result; because we can always
construct at least this series of proportional steps, beginning
with any one effective step~${\sr a}$, and proceeding
indefinitely in one direction:
$$1 \times {\sr a},\quad
      a \times {\sr a},\quad
      a \times a \times {\sr a},\ldots;
   \eqno (303.)$$
and we may still call the ratio~$1$ the {\it zero-power}, and the
ratios $a$,~$a \times a,\ldots$ the {\it positive powers\/} of
the ratio~$a$, even when we cannot continue this series of
proportional steps (303.) backward, like the series (259.), so as
to determine any contra-positive powers of $a$; namely, in that
particular case when $a = 0$.  We may, therefore, consider the
equation (264.), $a^0 = 1$, as including even this particular
case $a = 0$; and may write
$$0^0 = 1,
   \eqno (304.)$$
and, therefore, by (301.) and (295.)
$$0^{0 \over m} = 1:
   \eqno (305.)$$
we are also conducted to consider the symbols
$$0^{\oppos m},\quad 0^{\oppos n \over m},
   \eqno (306.)$$
as absurd, the ratio~$0$ having no contra-positive powers.

From the generality which we have been led to attribute to the
equation $a^0 = 1$, it follows that the symbol
$$1^{1 \over 0},
   \quad\hbox{and more generally}\quad 1^{\nu \over 0},
   \eqno (307.)$$
is indeterminate, or that it is equally fit to denote all ratios
whatever; but that the symbol
$$b^{1 \over 0},
   \quad\hbox{or}\quad b^{\nu \over 0},
   \quad\hbox{if}\quad b \neq 1,
   \eqno (308.)$$
is absurd, or that it cannot properly denote any ratio.  In
particular, the symbols
$$0^{1 \over 0},\quad 0^{\nu \over 0},
   \eqno (309.)$$
are absurd, or denote no ratios whatever.  In like manner the
symbol
$$0^{1 \over \oppos m},\quad 0^{\nu \over \oppos m},
   \eqno (310.)$$
is absurd, or denotes no ratio, because no ratio~$a$ can satisfy
the equation
$$a^{\oppos m} = 0.
   \eqno (311.)$$

We have thus discussed all the nine cases (298.), of powers of
which the base is $0$, and have found them all to be impossible,
except the two first, in which the exponents are
$\displaystyle {n \over m}$ and
$\displaystyle {0 \over m}$,
and in which the resulting powers are respectively $0$ or $1$.
We have also obtained sufficient elements for discussing all the
other cases (297.) and (299.), with their sub-divisions (300.),
as follows.

\smallskip

1st.
${\rm B}^{n \over m}$ is determined and positive, unless $m$ is
even, and $n$ odd; in which case it becomes of the form
${\rm B}^{\oppos 1 + 2k \over 2i}$, and is ambiguous, being
capable of denoting either of two opposite ratios, a positive or
a contra-positive.  To distinguish these among themselves, we may
denote the positive one by the symbol
$$\subzero{{\rm B}}^{\oppos 1 + 2k \over 2i},
   \eqno (312.)$$
and the contra-positive one by the symbol
$$\oppos \subzero{{\rm B}}^{\oppos 1 + 2k \over 2i};
   \eqno (313.)$$
for example, the two values of the square root
$\root 2 \of{\vphantom{{\rm B}}} {\rm B}$ or
${\rm B}^{1 \over 2}$,
may be denoted for distinction by the two separate symbols
$$\subzero{{\rm B}}^{1 \over 2} = \surd {\rm B},\quad
  \oppos \subzero{{\rm B}}^{1 \over 2} = \oppos \surd {\rm B}.
   \eqno (314.)$$
The other three cases of the notation ${\rm B}^{n \over m}$,
namely, the symbols
$${\rm B}^{\oppos 1 + 2k \over \oppos 1 + 2i},\quad
  {\rm B}^{2k \over \oppos 1 + 2i},\quad
  {\rm B}^{2k \over 2i},
   \eqno (315.)$$
denote determined positive ratios.

\smallskip

2nd.  The three cases
$${\rm B}^{1 + \oppos (2k) \over \oppos 1 + 2i},\quad
  {\rm B}^{\oppos (2k) \over \oppos 1 + 2i},\quad
  {\rm B}^{\oppos (2k) \over 2i},
   \eqno (316.)$$
of the notation ${\rm B}^{\oppos n \over m}$ are symbols of
determined positive ratios; but the case
${\rm B}^{1 + \oppos (2k) \over 2i}$ is ambiguous, this symbol
denoting either a determined positive ratio or a determined
contra-positive ratio, which may be thus respectively marked,
when we wish to distinguish them from each other,
$$\subzero{{\rm B}}^{1 + \oppos (2k) \over 2i},\quad
  \oppos \subzero{{\rm B}}^{1 + \oppos (2k) \over 2i}.
   \eqno (317.)$$
In general, we may write,
$${\rm B}^{\oppos n \over m} = \recip ({\rm B}^{n \over m}),
   \eqno (318.)$$
the latter of these two symbols having the same meaning or
meanings as the former.

\smallskip

3nd.  The symbols
$${\rm B}^{\oppos 1 + 2k \over 1 + \oppos (2i)},\quad
  {\rm B}^{2k \over 1 + \oppos (2i)},\quad
  {\rm B}^{2k \over \oppos (2i)},
   \eqno (319.)$$
included in the form ${\rm B}^{n \over \oppos m}$, denote
determined positive ratios; but the other symbol
${\rm B}^{\oppos 1 + 2k \over \oppos (2i)}$, included in the same
form ${\rm B}^{n \over \oppos m}$, is ambiguous, denoting either
a determined positive or a determined contra-positive ratio,
$$\subzero{{\rm B}}^{\oppos 1 + 2k \over \oppos (2i)},\quad
  \oppos \subzero{{\rm B}}^{\oppos 1 + 2k \over \oppos (2i)}.
   \eqno (320.)$$
In general, we may write
$${\rm B}^{n \over \oppos m} = \recip ({\rm B}^{n \over m}).
   \eqno (321.)$$

\smallskip

4th.  In like manner, we may write,
$${\rm B}^{\oppos n \over \oppos m} = {\rm B}^{n \over m},
   \eqno (322.)$$
the former symbol having always the same meaning or meanings as
the latter.  The cases
$${\rm B}^{1 + \oppos (2k) \over 1 + \oppos (2i)},\quad
  {\rm B}^{\oppos (2k) \over 1 + \oppos (2i)},\quad
  {\rm B}^{\oppos (2k) \over \oppos (2i)},
   \eqno (323.)$$
are symbols of determined positive ratios; but the case
${\rm B}^{1 + \oppos (2k) \over \oppos (2i)}$ is ambiguous, and
includes two opposite ratios, which may be thus respectively
denoted,
$$\subzero{{\rm B}}^{1 + \oppos (2k) \over \oppos (2i)},\quad
  \oppos \subzero{{\rm B}}^{1 + \oppos (2k) \over \oppos (2i)}.
   \eqno (324.)$$
In general, we shall denote by the symbol
$$\subzero{{\rm B}}^{\nu \over \mu},
   \quad\hbox{or}\quad \subzero{b}^{\nu \over \mu},
   \quad\hbox{if}\quad b > 0,\quad \mu \neq 0,\quad \nu \neq 0,
   \eqno (325.)$$
that positive ratio which is either the only value, or at least
one of the values of the symbol ${\rm B}^{\nu \over \mu}$ or
$b^{\nu \over \mu}$; and it is important to observe that this
positive ratio is not changed, when the form of the fractional
logarithm
$\displaystyle {\nu \over \mu}$
is changed, as if it were a fractional multiplier, by the rule
(135.), to the form
$\displaystyle {\omega \times \nu \over \omega \times \mu}$,
or (as it may be more concisely written)
$\displaystyle {\omega \nu \over \omega \mu}$;
that is,
$$\subzero{{\rm B}}^{\omega \nu \over \omega \mu}
   = \subzero{{\rm B}}^{\nu \over \mu}:
   \eqno (326.)$$
a theorem which is easily proved by means of the relation (268.),
combined with the determinateness (already proved) of that
positive ratio which results from powering or rooting any
proposed positive ratio by any positive or contra-positive whole
number.

\smallskip

5th.  With respect to the five remaining notations of the group
(297.), namely, those in which $0$ occurs, we have
$${\rm B}^{0 \over m} = 1;\quad {\rm B}^{0 \over \oppos m} = 1;
   \eqno (327.)$$
also the symbols
$${\rm B}^{n \over 0},\quad {\rm B}^{\oppos n \over 0},
   \eqno (328.)$$
are each indeterminate when ${\rm B} = 1$, and absurd in the
contrary case; and, finally, the symbol
$${\rm B}^{0 \over 0}
   \eqno (329.)$$
is absurd when ${\rm B} \neq 1$, but determined and $= 1$ when
${\rm B} = 1$.

\smallskip

6th.  Proceeding to the group (299.), the symbols
$$(\oppos {\rm B})^{n \over 0},\quad
  (\oppos {\rm B})^{0 \over 0},\quad
  (\oppos {\rm B})^{\oppos n \over 0},
   \eqno (330.)$$
are absurd; the symbols
$$(\oppos {\rm B})^{0 \over m},\quad
  (\oppos {\rm B})^{0 \over \oppos m},
   \eqno (331.)$$
are determined and each $=1$, if $m$ be odd, but otherwise they
are absurd; and the four remaining symbols
$$(\oppos {\rm B})^{n \over m},\quad
  (\oppos {\rm B})^{\oppos n \over m},\quad
  (\oppos {\rm B})^{n \over \oppos m},\quad
  (\oppos {\rm B})^{\oppos n \over \oppos m},
   \eqno (332.)$$
are absurd if $m$ be even, but denote determined positive
ratios when $m$ is odd, which ratios are positive if $n$ be even,
but contra-positive if $n$ be odd.

It must be remembered that all the foregoing discussion of the
cases of the general notation $b^{\nu \over \mu}$, for powers of
{\it fractional\/} logarithms, is founded on the definition laid
down in the 30th article, that $b^{\nu \over \mu}$ denotes the
$\nu$'th power of a $\mu$'th root of $b$, or in other words, the
$\nu$'th power of a ratio~$a$ of which $b$ is the $\mu$'th power.
When no such ratio~$a$ can be found, consistently with the
previous conception of powers with {\it integer\/} logarithms,
the symbol $b^{\nu \over \mu}$ is pronounced to be {\it absurd},
or to be incapable of denoting any ratio consistently with its
general definition; and when two or more such ratios~$a$ can be
found, each having its $\mu$'th power $= b$, we have pronounced
that the fractional power $b^{\nu \over \mu}$ is
{\it ambiguous\/} or {\it indeterminate}, except in those cases
in which the second component act of powering by the
numerator~$\nu$ has happened to destroy the indeterminateness.
And with respect to powers with {\it integer\/} exponents, it is
to be remembered that we always define them by a reference to a
series of proportional steps, of which at least the original step
(corresponding to the zero-power), is supposed to be an
{\it effective\/} step, and which can always be continued
indefinitely, at least in the positive direction, that is, in the
way of {\it repeated multiplication\/} by the ratio proposed as
the base, although in the particular case of a null ratio, we
cannot continue the series backward by {\it division}, so as to
find any contra-positive powers.  These definitions appear to be
the most natural; but others might have been assumed, and then
other results would have followed.  In general, the definitions
of mathematical science are not altogether arbitrary, but a
certain discretion is allowed in the selection of them, although
when once selected, they must then be consistently reasoned from.

\bigbreak

33.
The foregoing article enables us to assign one determined
positive ratio, and only one, as denoted by the symbol
$\subzero{b}^\alpha$, when $b$ is any determined positive ratio,
and $\alpha$ any fractional number with a numerator and a
denominator each different from $0$: it shows also that this
ratio $\subzero{b}^\alpha$ does not change when we transform the
expression of the fractional logarithm~$\alpha$ by introducing or
suppressing any whole number~$\omega$ as a factor common to both
numerator and denominator; and permits us to write
$$\subzero{b}^{\oppos \alpha} = \recip (\subzero{b}^\alpha),
   \eqno (333.)$$
$\oppos \alpha$ being the opposite of the fraction~$\alpha$ in
the sense of the 17th article.  More generally, by the meaning of
the notation $\subzero{b}^\alpha$, and by the determinateness of
those positive ratios which result from the powering or rooting
of determined positive ratios by determined integer numbers,
(setting aside the impossible of indeterminate case of rooting by
the number~$0$,) we have the relation
$$\subzero{b}^\beta \times \subzero{b}^\alpha
   =  \subzero{b}^{\beta + \alpha},
   \eqno (334.)$$
which is analogous to (267.); and the relation
$$\subzero{c}^\beta = \subzero{b}^{\beta \times \alpha}
   \quad\hbox{if}\quad c = \subzero{b}^\alpha,
   \eqno (335.)$$
analogous to (268.): $\alpha$ and $\beta$ denoting here any two
commensurable numbers.  And it is easy to see that while the
fractional exponent or logarithm~$\alpha$ increases, advancing
successively through all fractional states in the progression
from contra-positive to positive, the positive ratio
$\subzero{b}^\alpha$ increases constantly if $b > 1$, or else
decreases constantly if $b < 1$, ($b > 0$,) or remains constantly
$= 1$ if $b = 1$.  But to show that this increase or decrease of
the power with the exponent is {\it continuous\/} as well as
constant, we must establish principles for the interpretation of
the symbol $\subzero{b}^\alpha$ when $\alpha$ is not a fraction.

When $\alpha$ is incommensurable, but $b$ is still positive, it
may be proved that we shall still have these last relations
(334.) and (335.), if we interpret the symbol
$\subzero{b}^\alpha$ to denote that determined positive ratio~$c$
which satisfies the following conditions:
$$\left. \eqalign{
c = \subzero{b}^\alpha > \subzero{b}^{n' \over m'}
   &\quad\hbox{whenever}\quad \alpha > {n' \over m'},\cr
c = \subzero{b}^\alpha < \subzero{b}^{n'' \over m''}
   &\quad\hbox{whenever}\quad \alpha < {n'' \over m''},\cr
   &\quad\hbox{if}\quad b > 1;\cr}
   \right\}
   \eqno (336.)$$
or else these other conditions,
$$\left. \eqalign{
c = \subzero{b}^\alpha < \subzero{b}^{n' \over m'}
   &\quad\hbox{whenever}\quad \alpha > {n' \over m'},\cr
c = \subzero{b}^\alpha > \subzero{b}^{n'' \over m''}
   &\quad\hbox{whenever}\quad \alpha < {n'' \over m''},\cr
   &\quad\hbox{if}\quad b < 1,\quad b > 0;\cr}
   \right\}
   \eqno (337.)$$
or finally this equation,
$$c = \subzero{b}^\alpha = 1,
   \quad\hbox{if}\quad b = 1.
   \eqno (338.)$$
The reader will soon perceive the reasonableness of these
interpretations; but he may desire to see it proved that the
conditions (336.) or (337.) can always be satisfied by one
positive ratio~$c$, and only one, whatever determined ratio may
be denoted by $\alpha$, and whatever positive ratio (different
from~$1$) by $b$.  That {\it at least one\/} such positive ratio
$c = \subzero{b}^\alpha$ can be found, whatever incommensurable
number the exponent~$\alpha$ may be, is easily proved from the
circumstance that none of the conditions (336.) are incompatible
with one another if $b > 1$, and that none of the conditions
(337.) are incompatible with each other in the contrary case, by
reason of the constant increase or constant decrease of the
fractional power $\subzero{b}^{n \over m}$ for constantly
increasing values of the fractional exponent
$\displaystyle {n \over m}$.
And that {\it only one\/} such positive ratio
$c = \subzero{b}^\alpha$ can be found, or that no two different
positive ratios $c$, $c'$, can {\it both\/} satisfy {\it all\/}
these conditions may be proved for the case $b > 1$ by the
following process, which can without difficulty be adapted to the
other case.

The fractional powers of $b$ comprised in the series
$$\subzero{b}^{1 \over m},\quad
  \subzero{b}^{2 \over m},\quad
  \subzero{b}^{3 \over m},\quad\ldots\quad
  \subzero{b}^{im \over m},\quad
  \subzero{b}^{1 + im \over m},
   \eqno (339.)$$
increase (when $b > 1$) by increasing differences, of which the
last is
$$\oppos \subzero{b}^{im \over m}
       + \subzero{b}^{1 + im \over m}
   =   b^i ( \oppos 1 + \subzero{b}^{1 \over m} );
   \eqno (340.)$$
this last difference, therefore, and by still stronger reason
each of the others which precede it, will be less than
$\displaystyle {1 \over k}$, if
$$l > k b^i
   \eqno (341.)$$
and
$$\oppos 1 + \subzero{b}^{1 \over m} < {1 \over l}:
   \eqno (342.)$$
and this last condition will be satisfied, if
$$m > l ( \oppos 1 + b ),
   \eqno (343.)$$
$l$ and $m$ (like $i$ and $k$) denoting any positive whole
numbers; for then we shall have
$$1 + {m \over l} > b,
   \eqno (344.)$$
and by still stronger reason
$$\left( 1 \times {1 \over l} \right)^m > b,\quad
  1 + {1 \over l} > \subzero{b}^{1 \over m},
   \eqno (345.)$$
observing that
$$\left( 1 + {1 \over l} \right)^m > 1 + {m \over l},
   \quad\hbox{if}\quad m > 1,
   \eqno (346.)$$
because, by the theorem of multiplication (273.), or (281.),
$$\oppos 1 + \left( 1 + {1 \over l} \right)^m
   =  {1 \over l}
      \left\{
         1
          + \left( 1 + {1 \over l} \right)
          + \left( 1 + {1 \over l} \right)^2
          + \ldots
          + \left( 1 + {1 \over l} \right)^{\oppos 1 + m}
      \right\}.
   \eqno (347.)$$
If then $c$~$c'$ be any two proposed unequal positive ratios, of
which we may suppose that $c'$ is the greater,
$$c' > c,\quad c > 0,
   \eqno (348.)$$
we may choose two positive whole numbers $i$, $k$, so large that
$$b^i > c',\quad {1 \over k} < \oppos c + c',
   \eqno (349.)$$
and two other positive whole numbers $l$, $m$, large enough to
satisfy the conditions (341.) (343.); and then we shall be sure
that some one at least, such as $\subzero{b}^{n \over m}$, of the
fractional powers of $b$ comprised in the series (339.) will fall
between the two proposed unequal ratios $c$~$c'$, so that
$$c  < \subzero{b}^{n \over m},\quad
  c' > \subzero{b}^{n \over m}.
   \eqno (350.)$$
If then the one ratio~$c$ satisfy all the conditions (336.), the
incommensurable number~$\alpha$ must be
$\displaystyle < {n \over m}$,
and therefore, by the 2nd relation (350.), the other ratio~$c'$
cannot also satisfy all the conditions of the same form, since it
is
$\displaystyle > \subzero{b}^{n \over m}$,
although $\displaystyle \alpha < {n \over m}$.
In like manner, if the greater ratio~$c'$ satisfy all the
conditions of the form (336.) the lesser ratio~$c$ cannot also
satisfy them all, because in this case the incommensurable
number~$\alpha$ will be
$\displaystyle > {n \over m}$.
No two unequal positive ratios, therefore, can satisfy all those
conditions: they are therefore satisfied by one positive ratio
and only one, and the symbol $\subzero{b}^\alpha$ (interpreted
by them) denotes a determined positive ratio when $b > 1$.  For a
similar reason the same symbol $\subzero{b}^\alpha$, interpreted
by the conditions (337.), denotes a determined positive ratio
when $b < 1$, $b > 0$; and for the remaining case of a positive
base, $b = 1$, the symbol $\subzero{b}^\alpha$ denotes still, by
(338.) a determined positive ratio, namely, the ratio~$1$.  The
exponent or logarithm~$\alpha$ has, in these late investigations,
been supposed to be incommensurable; when that exponent~$\alpha$
is commensurable, the base~$b$ being still positive, we saw that
the symbol~$\subzero{b}^\alpha$ can be interpreted more easily,
as a power of a root, and that it always denotes a determined
positive ratio.

Reciprocally, in the equation
$$c = \subzero{b}^\alpha,
   \eqno (351.)$$
if the power~$c$ be any determined positive ratio, and if the
exponent~$\alpha$ be any determined ratio, positive or
contra-positive, we can deduce the positive base or ratio~$b$, by
calculating the inverse or reciprocal power
$$b = \subzero{c}^{\recip \alpha};
   \eqno (352.)$$
as appears from the relation (335.) which extends, as was above
announced, together with the relation (334.), even to the case of
incommensurable exponents.  The proof of the important extension
last alluded to, will easily suggest itself to those who have
studied the foregoing demonstrations; and they will perceive that
with the foregoing rules for the interpretation of the symbol
$\subzero{b}^\alpha$, for the case of an incommensurable
exponent, the power $\subzero{b}^\alpha$ increases (as was said
above) {\it continuously\/} as well as {\it constantly\/} with
the exponent~$\alpha$ if the base~$b$ be $> 1$, or else
decreases continuously and constantly if that positive base be
$< 1$, but remains constantly $= 1$ if $b = 1$.  It is therefore
possible to find one determined exponent or logarithm~$\alpha$
and only one, which shall satisfy the equation (351.), when the
power~$c$ and the base~$b$ are any given positive ratios, except
in the impossible or indeterminate case when this base~$b$ is the
particular ratio~$1$; and the number~$\alpha$ thus determined,
whether positive or contra-positive or null, may be called ``the
logarithm of $c$ to the base~$b$,'' and may be denoted by the
symbol
$$\alpha = \log_b \mathord{} \mathbin{.} c.
   \eqno (353.)$$

It is still more easy to perceive, finally, that when this
logarithm~$\alpha$ is given, (even if it be incommensurable,) the
power~$c$ increases constantly and continuously from zero with
the base~$b$, if $\alpha > 0$, or else decreases constantly and
continuously towards zero if $\alpha < 0$, or remains constant
and $= 1$, if $\alpha = 0$.

\bigbreak

\centerline{\it
Remarks on the Notation of this Essay, and on some modifications
by which it may be made}

\nobreak\vskip 3pt

\centerline{\it
more like the Notation commonly employed.}

\nobreak\bigskip

34.
In the foregoing articles we have constantly denoted
{\it moments}, or indivisible points of time, by small capital
letters, ${\sc a}$,~${\sc b}$,~${\sc a}'$,~${\sc b}'$, \&c.; and
{\it steps}, or transitions from one such moment to others, by
small Roman letters,
${\sr a}$,~${\sr b}$,~${\sr a}'$,~${\sr b}'$, \&c.
The mark~$-$ has been interposed between the marks of two
moments, to express the ordinal relation of those two moments, or
the step which must be made in order to pass from one to the
other; and the mark~$+$ has been inserted between the marks of a
step and a moment, or between the marks of two steps, to denote
the application of the step to the moment, or the composition of
the two steps with each other.  For the decomposition of a step
into others, we have used no special mark; but employed the
theorem that such decomposition can be performed by compounding
with the given compound step the opposites of the given component
steps, and a special notation for such opposite steps, namely the
mark~$\oppos$ prefixed; so that we have written $\oppos {\sr a}$
to denote the step opposite to the step~${\sr a}$, and
consequently $\oppos {\sr a} + {\sr b}$ to denote the algebraical
excess of the step~${\sr b}$ over the step~${\sr a}$, because
this excess may be conceived as a step compounded of ${\sr b}$
and $\oppos {\sr a}$.  However, we might have agreed to write
$$({\sr b} + {\sc a}) - ({\sr a} + {\sc a}) = {\sr b} - {\sr a},
   \eqno (354.)$$
denoting the step from the moment ${\sr a} + {\sc a}$ to the
moment ${\sr b} + {\sc a}$, for conciseness by
${\sr b} - {\sr a}$; and then ${\sr b} - {\sr a}$ would have been
another symbol for the algebraical excess of the step~${\sr b}$
over the step~${\sr a}$, and we should have had the equation
$${\sr b} - {\sr a} = \oppos {\sr a} + {\sr b}.
   \eqno (355.)$$
We might thus have been led to interpose the mark~$-$ between the
marks of a compound step~${\sr b}$ and a component
step~${\sr a}$, in order to denote the other component step, or
the algebraical {\it remainder\/} which results from the
algebraical {\it subtraction\/} of the component from the
compound.

Again, we have used the Greek letters
$\mu$~$\nu$~$\xi$~$\rho$~$\omega$, with or without accents, to
denote {\it integer numbers\/} in general, and the italic letters
$i$~$k$~$l$~$m$~$n$ to denote positive whole numbers in
particular; using also the earlier letters
$\alpha$~$\beta$~$\gamma$~$a$~$b$~$c$~$d$
to denote any ratios whatever, commensurable or incommensurable,
and in one recent investigation the capital letter~${\rm B}$ to
denote any positive ratio: and employed, in the combination of
these symbols of numbers, or of ratios, the same marks of
{\it addition\/} and of {\it opposition}, $+$ and $\oppos$, which
had been already employed for steps, and the mark of
multiplication~$\times$, without any special mark for
{\it subtraction}.  We might, however, have agreed to write, in
general,
$$(b \times {\sr a}) - (a \times {\sr a})
   =  (b - a) \times {\sr a},
   \eqno (356.)$$
as we wrote
$$(b \times {\sr a}) + (a \times {\sr a})
   =  (b + a) \times {\sr a};$$
and then the symbol $b - a$ would have denoted the algebraical
excess of the number~$b$ over the number~$a$, or the remainder
obtained by the algebraical {\it subtraction\/} of the latter
number from the former; and we should have had the equation,
$$b - a = \oppos a + b,
   \eqno (357.)$$
which is, with respect to {\it numbers}, or ratios, what the
equation (355.) is, with respect to steps.  And when such symbols
of remainders, ${\sr b} - {\sr a}$ or $b - a$, are to be combined
with other symbols in the way of algebraical {\it addition}, it
results, from principles already explained, that they need not be
enclosed in parentheses; for example, we may write simply
${\sr c} + {\sr b} - {\sr a}$ instead of
${\sr c} + ({\sr b} - {\sr a})$, because the sum denoted by this
last notation is equivalent to the remainder
$({\sr c} + {\sr b}) - {\sr a}$.  But the parentheses (or some
other combining mark) must be used, when a remainder is to be
{\it subtracted\/}; thus the symbol ${\sr c} - {\sr b} - {\sr a}$
is to be interpreted as $({\sr c} - {\sr b}) - {\sr a}$, and not
as ${\sr c} - ({\sr b} - {\sr a})$, which latter symbol is
equivalent to $({\sr c} - {\sr b}) + {\sr a}$, or
${\sr c} - {\sr b} + {\sr a}$.

\bigbreak

35.
With this way of denoting the algebraical subtraction of
steps, and that of numbers, we have the formul{\ae},
$$0 - {\sr a} = \oppos {\sr a},\quad
  0 - a = \oppos a,
   \eqno (358.)$$
$0$ denoting in the one a null step, and in the other a null
number.  And if we farther agree to suppress (for abridgement)
this symbol~$0$ which it occurs in such combinations as the
following, $0 + {\sr a}$, $0 - {\sr a}$, $0 + a$, $0 - a$,
writing, in the case of steps,
$$0 + {\sr a} = + {\sr a},\quad
  0 - {\sr a} = - {\sr a},
   \eqno (359.)$$
and similarly, in the case of numbers,
$$0 + a = + a,\quad
  0 - a = - a,
   \eqno (360.)$$
and, in like manner,
$$\left. \multieqalign{
0 + {\sr a} \pm {\sr b} &= + {\sr a} \pm {\sr b}, &
0 - {\sr a} \pm {\sr b} &= - {\sr a} \pm {\sr b}, \cr
0 + a \pm b &= + a \pm b, &
0 - a \pm b &= - a \pm b, \cr}
   \right\}
   \eqno (361.)$$
we shall then have the formul{\ae}
$$+ {\sr a} = {\sr a},\quad - {\sr a} = \oppos {\sr a},
   \eqno (362.)$$
and
$$+ a = a,\quad - a = \oppos a,
   \eqno (363.)$$
of which the one set refers to steps and the other to numbers.
With these conventions, the prefixing of the mark~$+$ to an
isolated symbol of a step or of a number, does not change the
meaning of the symbol; but the prefixing of the mark~$-$ converts
that symbol into another, which denotes the opposite of the
original step, or the opposite of the original number; so that
the series of whole numbers (103.) or (266.) may be written as
follows:
$$\ldots \, -3,\quad -2,\quad -1,\quad
  0,\quad +1,\quad +2,\quad +3,\ldots
   \eqno (364.)$$
Also, in this notation,
$$\left. \multieqalign{
{\sr b} \pm ( + {\sr a} )
   &= {\sr b} \pm {\sr a}, &
{\sr b} \pm ( - {\sr a} )
   &= {\sr b} \mp {\sr a},\cr
b \pm ( + a )
   &= b \pm a, &
b \pm ( - a )
   &= b \mp a.\cr}
   \right\}
   \eqno (365.)$$

\bigbreak

36.
Finally, as we wrote, for the case of commensurable steps,
$${\nu \times {\sr a} \over \mu \times {\sr a}}
  = {\nu \over \mu},$$
$\mu$ and $\nu$ being here whole numbers, so we may agree to
write, in general,
$${b \times {\sr a} \over a \times {\sr a}}
   = {b \over a},
   \eqno (366.)$$
whatever ratios $a$ and $b$ may be; and then this symbol
$\displaystyle {b \over a}$
will denote, in general, the algebraic quotient obtained by
dividing the number or ratio~$b$ by the number or ratio~$a$;
whereas we had before no general way of denoting such a quotient,
except by the mark~$\recip$ prefixed to the symbol of the
divisor~$a$, so as to form a symbol of the reciprocal
number~$\recip a$, to multiply by which latter number is
equivalent to dividing by the former.  Comparing the two
notations,
$${1 \over a} = \recip a,
   \eqno (367.)$$
and generally
$${b \over a} = \recip a \times b = b \times \recip a.
   \eqno (368.)$$
These two marks $\oppos$ and $\recip$ have been the only
{\it new\/} marks introduced in this Elementary Essay; although
the notation employed for powers differs a little from the common
notation: especially the symbol $\subzero{b}^\alpha$, suggested
by those researches of Mr.~Graves respecting the general
expression of powers and logarithms, which were the first
occasion of the conception of that Theory of Conjugate Functions
to which we now proceed.

\vfill\eject

\centerline{\largerm THEORY OF CONJUGATE FUNCTIONS,}

\vskip 6pt

\centerline{\largerm OR ALGEBRAIC COUPLES.}

\nobreak\bigskip

\centerline{\vbox{\hrule width 72pt}}

\nobreak\bigskip

\centerline{\it On Couples of Moments, and of Steps, in Time.}

\nobreak\bigskip

1.
When we have imagined any one moment of time~${\sc a}_1$, which
we may call a {\it primary moment}, we may again imagine a moment
of time~${\sc a}_2$, and may call this a {\it secondary moment},
without regarding whether it follows, or coincides with, or
precedes the primary, in the common progression of time; we may
also speak of this primary and this secondary moment as forming a
{\it couple of moments}, or a {\it moment-couple}, which may be
denoted thus, $({\sc a}_1, {\sc a}_2)$.  Again, we may imagine
any other two moments, a primary and a secondary, ${\sc b}_1$ and
${\sc b}_2$, distinct from or coincident with each other, and
forming another {\it moment-couple}, $({\sc b}_1, {\sc b}_2)$;
and we may compare the latter couple of moments with the former,
moment with moment, primary with primary, and secondary with
secondary, examining how ${\sc b}_1$ is ordinally related to
${\sc a}_1$, and how ${\sc b}_2$ is ordinally related to
${\sc a}_2$, in the progression of time, as coincident, or
subsequent, or precedent; and thus may obtain a {\it couple of
ordinal relations}, which may be thus separately denoted
${\sc b}_1 - {\sc a}_1$, ${\sc b}_2 - {\sc a}_2$, or thus
collectively, as a {\it relation-couple},
$$({\sc b}_1 - {\sc a}_1, {\sc b}_2 - {\sc a}_2).$$
This couple of ordinal relations between moments may also be
conceived as constituting a complex {\it relation of one
moment-couple to another\/}; and in conformity with this
conception it may be thus denoted,
$$({\sc b}_1, {\sc b}_2) - ({\sc a}_1, {\sc a}_2),$$
so that, comparing this with the former way of representing it,
we may establish the written equation,
$$({\sc b}_1, {\sc b}_2) - ({\sc a}_1, {\sc a}_2)
   =  ({\sc b}_1 - {\sc a}_1, {\sc b}_2 - {\sc a}_2).
   \eqno (1.)$$

Instead of conceiving thus a couple of ordinal relations between
moments, or a relation between two couples of moments, discovered
by the (analytic) {\it comparison\/} of one such couple of
moments with another, we may conceive a {\it couple of steps\/}
in the progression of time, from moment to moment respectively,
or a single complex step which we may call a {\it step-couple\/}
from one moment-couple to another, serving to {\it generate\/}
(synthetically) one of these moment-couples from the other; and
if we denote the two separate steps by ${\sr a}_1$,~${\sr a}_2$,
(${\sr a}_1$ being the step from ${\sc a}_1$ to ${\sc b}_1$, and
${\sr a}_2$ being the step from ${\sc a}_2$ to ${\sc b}_2$,) so
that in the notation of the Preliminary Essay,
$${\sc b}_1 = {\sr a}_1 + {\sc a}_1,\quad
  {\sc b}_2 = {\sr a}_2 + {\sc a}_2,$$
$${\sc b}_1 = ({\sc b}_1 - {\sc a}_1) + {\sc a}_1,\quad
  {\sc b}_2 = ({\sc b}_2 - {\sc a}_2) + {\sc a}_2,$$
we may now establish this analogous notation for couples,
$$\left. \eqalign{
({\sc b}_1, {\sc b}_2)
   &= ({\sr a}_1 + {\sc a}_1, {\sr a}_2 + {\sc a}_2) \cr
   &= ({\sr a}_1, {\sr a}_2) + ({\sc a}_1, {\sc a}_2) \cr
   &= \{ ({\sc b}_1, {\sc b}_2) - ({\sc a}_1, {\sc a}_2) \}
       + ({\sc a}_1, {\sc a}_2),\cr}
   \right\}
   \eqno (2.)$$
the symbol $({\sc b}_1, {\sc b}_2) - ({\sc a}_1, {\sc a}_2)$
corresponding now to the conception of the {\it step-couple\/} by
which we may pass from the {\it moment-couple\/} $({\sc a}_1,
{\sc a}_2)$ to the {\it moment-couple\/}
$({\sc b}_1, {\sc b}_2)$, and the equivalent symbol
$({\sr a}_1, {\sr a}_2)$ or
$({\sc b}_1 - {\sc a}_1, {\sc b}_2 - {\sc a}_2)$ corresponding
now to the conception of the {\it couple of steps\/}
${\sr a}_1$,~${\sr a}_2$, from the two moments
${\sc a}_1$,~${\sc a}_2$, to the two moments
${\sc b}_1$,~${\sc b}_2$, respectively.  The step~${\sr a}_1$, or
${\sc b}_1 - {\sc a}_1$ may be called the {\it primary step\/} of
the couple $({\sr a}_1, {\sr a}_2)$, and the step~${\sr a}_2$ or
${\sc b}_2 - {\sc a}_2$ may be called the {\it secondary step}.

A step-couple may be said to be {\it effective\/} when it
actually changes the moment-couple to which it is applied; that
is, when one at least of its two coupled steps is effective: and
in the contrary case, that is, when both those coupled steps are
separately null, the step-couple itself may be said to be
{\it null\/} also.  And effective step-couples may be
distinguished into {\it singly effective\/} and {\it doubly
effective\/} step-couples, according as they alter {\it one\/} or
{\it both\/} of the two moments of the moment-couples to which
they are applied.  Finally, a singly effective step-couple may be
called a {\it pure primary\/} or {\it pure secondary\/}
step-couple, according as only its primary or only its secondary
step is effective, that is, according as it alters only the
primary or only the secondary moment.  Thus $(0,0)$ is a null
step-couple, $({\sr a}_1, {\sr a}_2)$ is a doubly effective
step-couple, and $({\sr a}_1, 0)$ $(0, {\sr a}_2)$ are
singly effective step-couples, the former $({\sr a}_1, 0)$ being
a pure primary, and the latter $(0, {\sr a}_2)$ being a pure
secondary, if $0$ denote a null step and ${\sr a}_1$, ${\sr a}_2$
effective steps.

\bigbreak

\centerline{\it
On the Composition and Decomposition of Step-Couples.}

\nobreak\bigskip

2.
Having stepped from one couple of moments
$({\sc a}_1, {\sc a}_2)$ to another couple of moments
$({\sc b}_1, {\sc b}_2)$ by one step-couple
$({\sr a}_1, {\sr a}_2)$, we may afterwards step
to a third couple of moments $({\sc c}_1, {\sc c}_2)$ by a second
step-couple $({\sr b}_1, {\sr b}_2)$, so as to have
$$\left. \eqalign{
({\sc c}_1, {\sc c}_2)
   &= ({\sr b}_1, {\sr b}_2) + ({\sc b}_1, {\sc b}_2),\cr
({\sc b}_1, {\sc b}_2)
   &= ({\sr a}_1, {\sr a}_2) + ({\sc a}_1, {\sc a}_2);\cr}
   \right\}
   \eqno (3.)$$
and then we may consider ourselves as having made upon the whole
a compound couple of steps, or a {\it compound step-couple}, from
the first moment-couple $({\sc a}_1, {\sc a}_2)$ to the third
moment-couple $({\sc c}_1, {\sc c}_2)$, and may agree to call
this compound step-couple the {\it sum\/} of the two component
step-couples
$({\sr a}_1, {\sr a}_2)$, $({\sr b}_1, {\sr b}_2)$,
or to say that it is formed by {\it adding\/} them, and to denote
it as follows,
$$({\sc c}_1, {\sc c}_2) - ({\sc a}_1, {\sc a}_2)
   =  ({\sr b}_1, {\sr b}_2) + ({\sr a}_1, {\sr a}_2);
   \eqno (4.)$$
as, in the language of the Preliminary Essay, the two separate
compound steps, from ${\sc a}_1$ to ${\sc c}_1$ and from
${\sc a}_2$ to ${\sc c}_2$ are the {\it sums\/} of the component
steps, and are denoted by the symbols ${\sr b}_1 + {\sr a}_1$ and
${\sr b}_2 + {\sr a}_2$ respectively.  With these notations, we
have evidently the equation
$$({\sr b}_1, {\sr b}_2) + ({\sr a}_1, {\sr a}_2)
   =  ({\sr b}_1 + {\sr a}_1, {\sr b}_2 + {\sr a}_2);
   \eqno (5.)$$
that is, the {\it sum of two step-couples may be formed by
coupling the two sum-steps}.  Hence, also,
$$({\sr b}_1, {\sr b}_2) + ({\sr a}_1, {\sr a}_2)
   =  ({\sr a}_1, {\sr a}_2) + ({\sr b}_1, {\sr b}_2),
   \eqno (6.)$$
that is, {\it the order of any two component step-couples may be
changed without altering the result\/}; and
$$({\sr a}_1, {\sr a}_2)
   =  ({\sr a}_1, 0) + (0, {\sr a}_2),
   \eqno (7.)$$
that is, {\it every doubly effective step-couple is the sum of a
pure primary and a pure secondary}.  In like manner, we can
conceive sums of more than two step-couples, and may establish,
for such sums, relations analogous to those marked (5.) and (6.);
thus,
$$\left. \eqalign{
         ({\sr c}_1, {\sr c}_2)
       + ({\sr b}_1, {\sr b}_2)
       + ({\sr a}_1, {\sr a}_2)
   &= (  {\sr c}_1 + {\sr b}_1 + {\sr a}_1,
         {\sr c}_2 + {\sr b}_2 + {\sr a}_2 ) \cr
   &=    ({\sr a}_1, {\sr a}_2)
       + ({\sr b}_1, {\sr b}_2)
       + ({\sr c}_1, {\sr c}_2)
         \quad\hbox{\&c.} \cr}
   \right\}
   \eqno (8.)$$

We may also consider the {\it decomposition\/} of a step-couple,
or the {\it subtraction\/} of one such step-couple from another,
and may write,
$$({\sr b}_1, {\sr b}_2) - ({\sr a}_1, {\sr a}_2)
   =  ({\sr b}_1 - {\sr a}_1, {\sr b}_2 - {\sr a}_2),
   \eqno (9.)$$
$({\sr b}_1, {\sr b}_2) - ({\sr a}_1, {\sr a}_2)$ being that
sought step-couple which must be compounded with or added to the
given component step-couple $({\sr a}_1, {\sr a}_2)$ in order to
produce the given compound step-couple $({\sr b}_1, {\sr b}_2)$.
And if we agree to suppress the symbol of a null step-couple,
when it is combined with others or others with it in the way of
addition or subtraction, we may write
$$\left. \eqalign{
({\sr a}_1, {\sr a}_2)
   &= (0, 0) + ({\sr a}_1, {\sr a}_2)
    =  + ({\sr a}_1, {\sr a}_2),\cr
(-{\sr a}_1, -{\sr a}_2)
   &= (0, 0) - ({\sr a}_1, {\sr a}_2)
    =  - ({\sr a}_1, {\sr a}_2),\cr}
   \right\}
   \eqno (10.)$$
employing a notation analogous to that explained for single steps
in the 35th article of the Preliminary Essay.  Thus
$+ ({\sr a}_1, {\sr a}_2)$ is another way of denoting the
step-couple $({\sr a}_1, {\sr a}_2)$ itself; and
$- ({\sr a}_1, {\sr a}_2)$ is a way of denoting the
{\it opposite\/} step-couple $(- {\sr a}_1, - {\sr a}_2)$.

\bigbreak

\centerline{\it
On the Multiplication of a Step-Couple by a Number.}

\nobreak\bigskip

3.
From any proposed moment-couple $({\sc a}_1, {\sc a}_2)$, and any
proposed step-couple $({\sr a}_1, {\sr a}_2)$, we may generate a
series of other moment-couples
$$\ldots \,
   ({\sc e}_1', {\sc e}_2'),\quad
   ({\sc e}_1, {\sc e}_2),\quad
   ({\sc a}_1, {\sc a}_2),\quad
   ({\sc b}_1, {\sc b}_2),\quad
   ({\sc b}_1', {\sc b}_2') \, \ldots
   \eqno (11.)$$
by repeatedly applying this step-couple $({\sr a}_1, {\sr a}_2)$
itself, and the opposite step-couple $- ({\sr a}_1, {\sr a}_2)$,
or $(- {\sr a}_1, - {\sr a}_2)$, in a way analogous to the
process of the 13th article of the Preliminary Essay, as follows:
$$\left. \eqalign{
\noalign{\vbox{\hbox{$\cdots\cdots$}}}
({\sc e}_1', {\sc e}_2')
   &=    (- {\sr a}_1, - {\sr a}_2)
       + (- {\sr a}_1, - {\sr a}_2)
       + ({\sc a}_1, {\sc a}_2),\cr
({\sc e}_1, {\sc e}_2)
   &=    (- {\sr a}_1, - {\sr a}_2)
       + ({\sc a}_1, {\sc a}_2),\cr
({\sc a}_1, {\sc a}_2)
   &=    ({\sc a}_1, {\sc a}_2),\cr
({\sc b}_1, {\sc b}_2)
   &=    ({\sr a}_1, {\sr a}_2)
       + ({\sc a}_1, {\sc a}_2),\cr
({\sc b}_1', {\sc b}_2')
   &=    ({\sr a}_1, {\sr a}_2)
       + ({\sr a}_1, {\sr a}_2)
       + ({\sc a}_1, {\sc a}_2),\cr
\noalign{\vbox{\hbox{$\cdots\cdots$}}}}
   \right\}
   \eqno (12.)$$
and a series of {\it multiple step-couples}, namely
$$\left. \eqalign{
\noalign{\vbox{\hbox{$\cdots\cdots$}}}
({\sc e}_1', {\sc e}_2') - ({\sc a}_1, {\sc a}_2)
   &=    (- {\sr a}_1, - {\sr a}_2)
       + (- {\sr a}_1, - {\sr a}_2),\cr
({\sc e}_1, {\sc e}_2) - ({\sc a}_1, {\sc a}_2)
   &=    (- {\sr a}_1, - {\sr a}_2),\cr
({\sc a}_1, {\sc a}_2) - ({\sc a}_1, {\sc a}_2)
   &=    (0, 0),\cr
({\sc b}_1, {\sc b}_2) - ({\sc a}_1, {\sc a}_2)
   &=    ({\sr a}_1, {\sr a}_2),\cr
({\sc b}_1', {\sc b}_2') - ({\sc a}_1, {\sc a}_2)
   &=    ({\sr a}_1, {\sr a}_2)
       + ({\sr a}_1, {\sr a}_2),\cr
\noalign{\vbox{\hbox{$\cdots\cdots$}}}}
   \right\}
   \eqno (13.)$$
which may be thus more concisely denoted,
$$\left. \eqalign{
\noalign{\vbox{\hbox{$\cdots\cdots$}}}
({\sc e}_1', {\sc e}_2')
   &=  - 2 ({\sr a}_1, {\sr a}_2)
       + ({\sc a}_1, {\sc a}_2),\cr
({\sc e}_1, {\sc e}_2)
   &=  - 1 ({\sr a}_1, {\sr a}_2)
       + ({\sc a}_1, {\sc a}_2),\cr
({\sc a}_1, {\sc a}_2)
   &=  \mathrel{\phantom{+}}
         0 ({\sr a}_1, {\sr a}_2)
       + ({\sc a}_1, {\sc a}_2),\cr
({\sc b}_1, {\sc b}_2)
   &=  + 1 ({\sr a}_1, {\sr a}_2)
       + ({\sc a}_1, {\sc a}_2),\cr
({\sc b}_1', {\sc b}_2')
   &=  + 2 ({\sr a}_1, {\sr a}_2)
       + ({\sc a}_1, {\sc a}_2),\cr
\noalign{\vbox{\hbox{$\cdots\cdots$}}}}
   \right\}
   \eqno (14.)$$
and
$$\left. \eqalign{
\noalign{\vbox{\hbox{$\cdots\cdots$}}}
({\sc e}_1', {\sc e}_2') - ({\sc a}_1, {\sc a}_2)
   &=  - 2 ({\sr a}_1, {\sr a}_2)
    =  - 2 \times ({\sr a}_1, {\sr a}_2),\cr
({\sc e}_1, {\sc e}_2) - ({\sc a}_1, {\sc a}_2)
   &=  - 1 ({\sr a}_1, {\sr a}_2)
    =  - 1 \times ({\sr a}_1, {\sr a}_2),\cr
({\sc a}_1, {\sc a}_2) - ({\sc a}_1, {\sc a}_2)
   &=  \mathrel{\phantom{+}}
         0 ({\sr a}_1, {\sr a}_2)
    =  \mathrel{\phantom{+}}
         0 \times ({\sr a}_1, {\sr a}_2),\cr
({\sc b}_1, {\sc b}_2) - ({\sc a}_1, {\sc a}_2)
   &=  + 1 ({\sr a}_1, {\sr a}_2)
    =  + 1 \times ({\sr a}_1, {\sr a}_2),\cr
({\sc b}_1', {\sc b}_2') - ({\sc a}_1, {\sc a}_2)
   &=  + 2 ({\sr a}_1, {\sr a}_2)
    =  + 2 \times ({\sr a}_1, {\sr a}_2),\cr
\noalign{\vbox{\hbox{\&c.}}}}
   \right\}
   \eqno (15.)$$
We may also conceive step-couples which shall be
{\it sub-multiples\/} and {\it fractions\/} of a given
step-couple, and may write
$$({\sr c}_1, {\sr c}_2)
   =  {\nu \over \mu} \times ({\sr b}_1, {\sr b}_2)
   =  {\nu \over \mu} ({\sr b}_1, {\sr b}_2),
   \eqno (16.)$$
when the two step-couples $({\sr b}_1, {\sr b}_2)$
$({\sr c}_1, {\sr c}_2)$ are related as multiples to one common
step-couple $({\sr a}_1, {\sr a}_2)$ as follows:
$$({\sr b}_1, {\sr b}_2)
   =  \mu \times ({\sr a}_1, {\sr a}_2),\quad
  ({\sr c}_1, {\sr c}_2)
   =  \nu \times ({\sr a}_1, {\sr a}_2),
   \eqno (17.)$$
$\mu$ and $\nu$ being any two proposed whole numbers.  And if we
suppose the {\it fractional multiplier\/}
$\displaystyle {\nu \over \mu}$
in (16.) to tend to any {\it incommensurable limit\/}~$a$, we may
denote by $a \times ({\sr b}_1, {\sr b}_2)$ the corresponding
limit of the fractional product, and may consider this latter
limit as the {\it product\/} obtained by multiplying the
step-couple $({\sr b}_1, {\sr b}_2)$ by the {\it incommensurable
multiplier\/} or number~$a$; so that we may write,
$$\left. \eqalign{
({\sr c}_1, {\sr c}_2)
   &= a \times ({\sr b}_1, {\sr b}_2)
    = a ({\sr b}_1, {\sr b}_2),\cr
\hbox{if}\quad
({\sr c}_1, {\sr c}_2)
   &= \underline{\rm L}
         \left( {\nu \over \mu} ({\sr b}_1, {\sr b}_2) \right)
   \quad\hbox{and}\quad
      a = \underline{\rm L} {\nu \over \mu},\cr}
   \right\}
   \eqno (18.)$$
using $\underline{\rm L}$ as the mark of a limit, as in the
notation of the Preliminary Essay.  It follows from these
conceptions of the multiplication of a step-couple by a number,
that generally
$$a \times ({\sr a}_1, {\sr a}_2)
   =  (a {\sr a}_1, a {\sr a}_2),
   \eqno (19.)$$
whatever steps may be denoted by ${\sr a}_1$,~${\sr a}_2$, and
whatever number (commensurable or incommensurable, and positive
or contra-positive or null) may be denoted by $a$.  Hence also we
may write
$${ (a {\sr a}_1, a {\sr a}_2) \over ({\sr a}_1, {\sr a}_2) }
   =  a,
   \eqno (20.)$$
and may consider the number~$a$ as expressing the {\it ratio\/}
of the step-couple $(a {\sr a}_1, a {\sr a}_2)$ to the
step-couple $({\sr a}_1, {\sr a}_2)$.

\bigbreak

\centerline{\it
On the Multiplication of a Step-Couple by a Number-Couple; and on
the Ratio of one}

\nobreak\vskip 3pt

\centerline{\it
Step-Couple to another.}

\nobreak\bigskip

4.
The formula (20.) enables us, in an infinite variety of cases, to
assign a single number~$a$ as the ratio of one proposed
step-couple $({\sr b}_1, {\sr b}_2)$ to another
$({\sr a}_1, {\sr a}_2)$; namely, in all those cases in which the
primary and secondary steps of the one couple are proportional to
those of the other: but it fails to assign such a ratio, in all
those cases in which this condition is not satisfied.  The spirit
of the present Theory of Couples leads us, however, to conceive
that the ratio of any one effective step-couple to any other may
perhaps be expressed in general by a {\it number-couple}, or
couple of numbers, a primary and a secondary; and that with
reference to this more general view of such ratio, the relation
(20.) might be more fully written thus,
$${ (a_1 {\sr a}_1, a_1 {\sr a}_2) \over ({\sr a}_1, {\sr a}_2) }
   =  (a_1, 0),
   \eqno (21.)$$
and the relation (19.) as follows,
$$(a_1, 0) \times ({\sr a}_1, {\sr a}_2)
   =  (a_1, 0) ({\sr a}_1, {\sr a}_2)
   =  (a_1 {\sr a}_1, a_1 {\sr a}_2),
   \eqno (22.)$$
the single number~$a_1$ being changed to the couple $(a_1, 0)$,
which may be called a {\it pure primary number-couple}.  The
spirit of this theory of primaries and secondaries leads us also
to conceive that the ratio of any step-couple
$({\sr b}_1, {\sr b}_2)$ to any pure primary step-couple
$({\sr a}_1, 0)$, may be expressed by coupling the two ratios
$\displaystyle {{\sr b}_1 \over {\sr a}_1}$,
$\displaystyle {{\sr b}_2 \over {\sr a}_1}$,
which the two steps ${\sr b}_1$, ${\sr b}_2$ bear to the
effective primary step~${\sr a}_1$; so that we may write
$${ ({\sr b}_1, {\sr b}_2) \over ({\sr a}_1, 0) }
   =  \left(
         {{\sr b}_1 \over {\sr a}_1},
         {{\sr b}_2 \over {\sr a}_1}
      \right),\quad
  { (a_1 {\sr a}_1, a_2 {\sr a}_1) \over ({\sr a}_1, 0) }
   =  (a_1, a_2),
   \eqno (23.)$$
and in like manner, by the general connexion of multiplication
with ratio,
$$(a_1, a_2) \times ({\sr a}_1, 0)
   =  (a_1, a_2) ({\sr a}_1, 0)
   =  (a_1 {\sr a}_1, a_2 {\sr a}_1).
   \eqno (24.)$$
From the relations (22.) (24.), it follows by (5.) that
$$(b_1 + a_1, 0) ({\sr a}_1, {\sr a}_2)
   =  (b_1, 0) ({\sr a}_1, {\sr a}_2)
    + (a_1, 0) ({\sr a}_1, {\sr a}_2),
   \eqno (25.)$$
and that
$$(a_1, a_2) ({\sr b}_1 + {\sr a}_1, 0)
   =  (a_1, a_2) ({\sr b}_1, 0)
    + (a_1, a_2) ({\sr a}_1, 0);
   \eqno (26.)$$
and the spirit of the present extension of reasonings and
operations on single moments, steps, and numbers, to
moment-couples, step-couples, and number-couples, leads us to
determine (if we can) what remains yet undetermined in the
conception of a number-couple, as a multiplier or as a ratio, so
as to satisfy the two following more general conditions,
$$(b_1 + a_1, b_2 + a_2) ({\sr a}_1, {\sr a}_2)
   =   (b_1, b_2) ({\sr a}_1, {\sr a}_2)
     + (a_1, a_2) ({\sr a}_1, {\sr a}_2),
   \eqno (27.)$$
and
$$(a_1, a_2) ({\sr b}_1 + {\sr a}_1, {\sr b}_2 + {\sr a}_2)
   =   (a_1, a_2) ({\sr b}_1, {\sr b}_2)
     + (a_1, a_2) ({\sr a}_1, {\sr a}_2),
   \eqno (28.)$$
whatever numbers may be denoted by $a_1$~$a_2$~$b_1$~$b_2$, and
whatever steps by
${\sr a}_1$~${\sr a}_2$~${\sr b}_1$~${\sr b}_2$.  With these
conditions we have
$$\eqalignno{
(a_1, a_2) ({\sr a}_1, {\sr a}_2)
   &=    (a_1, 0) ({\sr a}_1, {\sr a}_2)
       + (0, a_2) ({\sr a}_1, {\sr a}_2), &(29.)\cr
(0, a_2) ({\sr a}_1, {\sr a}_2)
   &=    (0, a_2) ({\sr a}_1, 0)
       + (0, a_2) (0, {\sr a}_2), &(30.)\cr}$$
and therefore, by (22.) and (24.), and by the formula for sums,
$$\eqalignno{
(a_1, a_2) ({\sr a}_1, {\sr a}_2)
   &=    (a_1 {\sr a}_1, a_1 {\sr a}_2)
       + (0, a_2 {\sr a}_1) + (0, a_2) (0,  {\sr a}_2) \cr
   &=    (a_1 {\sr a}_1, a_1 {\sr a}_2 + a_2 {\sr a}_1)
       + (0, a_2) (0,  {\sr a}_2), &(31.)\cr}$$
in which the product $(0, a_2) (0, {\sr a}_2)$ remains still
undetermined.  It must, however, by the spirit of the present
theory, be supposed to be some step-couple,
$$(0, a_2) (0, {\sr a}_2) = ({\sr c}_1, {\sr c}_2);
   \eqno (32.)$$
and these two steps ${\sr c}_1$~${\sr c}_2$ must each vary
proportionally to the product $a_2 {\sr a}_2$, since otherwise it
could be proved that the foregoing conditions (27.) and (28.)
would not be satisfied; we are, therefore, to suppose
$${\rm c_1} = \gamma_1 a_2 {\sr a}_2,\quad
  {\rm c_2} = \gamma_2 a_2 {\sr a}_2,
   \eqno (33.)$$
that is,
$$(0, a_2) (0, {\sr a}_2)
   =  (\gamma_1 a_2 {\sr a}_2, \gamma_2 a_2 {\sr a}_2),
   \eqno (34.)$$
$\gamma_1$, $\gamma_2$, being some two constant numbers,
independent of $a_2$ and ${\sr a}_2$, but otherwise capable of
being chosen at pleasure.  Thus, the general formula for the
product of a step-couple $({\sr a}_1, {\sr a}_2)$ multiplied by a
number-couple $(a_1, a_2)$, is, by (31.) (34.) and by the theorem
for sums,
$$\eqalignno{
(a_1, a_2) ({\sr a}_1, {\sr a}_2)
   &=    (a_1 {\sr a}_1, a_1 {\sr a}_2 + a_2 {\sr a}_1)
       + (\gamma_1 a_2 {\sr a}_2, \gamma_2 a_2 {\sr a}_2) \cr
   &=    (a_1 {\sr a}_1 + \gamma_1 a_1 {\sr a}_2,
          a_1 {\sr a}_2 + a_2 {\sr a}_1 + \gamma_2 a_2 {\sr a}_2):
      &(35.)\cr}$$
and accordingly this formula satisfies the conditions (27.) and
(28.), and includes the relations (22.) and (24.), whatever
arbitrary numbers we choose for $\gamma_1$, and $\gamma_2$;
provided that after once choosing these numbers, which we may
call the {\it constants of multiplication}, we retain them
thenceforth unaltered, and treat them as independent of both the
multiplier and the multiplicand.  It is clear, however, that the
simplicity and elegance of our future operations and results must
mainly depend on our making a simple and suitable choice of these
two constants of multiplication; and that in making this choice,
we ought to take care to satisfy, if possible, the essential
condition that there shall be always {\it one determined
number-couple to express the ratio of any one determined
step-couple to any other}, at least when the latter is not null:
since this was the very object, to accomplish which we were led
to introduce the conception of these number-couples.  It is easy
to show that no choice simpler than the following,
$$\gamma_1 = -1,\quad \gamma_2 = 0,
   \eqno (36.)$$
would satisfy this essential condition: and for that reason we
shall now select these two numbers, contra-positive one and zero,
for the two constants of multiplication, and shall establish
finally this formula for the multiplication of any step-couple
$({\sr a}_1, {\sr a}_2)$ by any number-couple $(a_1, a_2)$,
$$(a_1, a_2) ({\sr a}_1, {\sr a}_2)
   =  ( a_1 {\sr a}_1 - a_2 {\sr a}_2,
        a_2 {\sr a}_1 + a_1 {\sr a}_2 ).
   \eqno (37.)$$

\bigbreak

5.
In fact, whatever constants of multiplication
$\gamma_1$,~$\gamma_2$ we may select, if we denote by
$({\sr b}_1, {\sr b}_2)$ the product of the step-couple
$({\sr a}_1, {\sr a}_2)$ by the number-couple $(a_1, a_2)$, so
that
$$({\sr b}_1, {\sr b}_2)
   =  (a_1, a_2) \times ({\sr a}_1, {\sr a}_2),
   \eqno (38.)$$
we have by (35.) the following expressions for the two coupled
steps ${\sr b}_1$,~${\sr b}_2$, of the product,
$$\left. \eqalign{
{\sr b}_1
   &= a_1 {\sr a}_1
      + \gamma_1 a_2 {\sr a}_2,\cr
{\sr b}_2
   &= a_1 {\sr a}_2 + a_2 {\sr a}_1
      + \gamma_2 a_2 {\sr a}_2,\cr}
   \right\}
   \eqno (39.)$$
and therefore
$$\left. \eqalign{
\beta_1
   &= a_1 \alpha_1
      + \gamma_1 a_2 \alpha_2,\cr
\beta_2
   &= a_1 \alpha_2 + a_2 \alpha_1
      + \gamma_2 a_2 \alpha_2,\cr}
   \right\}
   \eqno (40.)$$
if $\alpha_1$~$\alpha_2$~$\beta_1$~$\beta_2$ denote respectively
the ratios of the four steps
${\sr a}_1$~${\sr a}_2$~${\sr b}_1$~${\sr b}_2$
to one effective step~${\sr c}$, so that
$${\sr a}_1 = \alpha_1 {\sr c},\quad
  {\sr a}_2 = \alpha_2 {\sr c},
   \eqno (41.)$$
and
$${\sr b}_1 = \beta_1 {\sr c},\quad
  {\sr b}_2 = \beta_2 {\sr c};
   \eqno (42.)$$
from which it follows that
$$\left. \eqalign{
a_1 \{ \alpha_1 (\alpha_1 + \gamma_2 \alpha_2)
        - \gamma_1 \alpha_2^2 \}
   &=    \beta_1 (\alpha_1 + \gamma_2 \alpha_2)
       - \beta_2 \gamma_1 \alpha_2,\cr
a_2 \{ \alpha_1 (\alpha_1 + \gamma_2 \alpha_2)
        - \gamma_1 \alpha_2^2 \}
   &=    \beta_2 \alpha_1 - \beta_1 \alpha_2;\cr}
   \right\}
   \eqno (43.)$$
in order therefore that the numbers $a_1$,~$a_2$ should always be
determined by the equation~(38.), when ${\sr a}_1$ and
${\sr a}_2$ are not both null steps, it is necessary and
sufficient that the factor
$$\alpha_1 (\alpha_1 + \gamma_2 \alpha_2)
         - \gamma_1 \alpha_2^2
   =  (\alpha_1
         + {\textstyle {1 \over 2}} \gamma_2 \alpha_2)^2
    - (\gamma_1
         + {\textstyle {1 \over 4}} \gamma_2^2) \alpha_2^2
   \eqno (44.)$$
should never become null, when $\alpha_1$ and $\alpha_2$ are not
both null numbers; and this condition will be satisfied if we so
choose the constants of multiplication $\gamma_1$,~$\gamma_2$ as
to make
$$\gamma_1 + {\textstyle {1 \over 4}} \gamma_2^2 < 0,
   \eqno (45.)$$
but not otherwise.  Whatever constants $\gamma_1$,~$\gamma_2$ we
choose, we have, by the foregoing principles,
$${({\sr c}, 0) \over ({\sr c}, 0)} = (1, 0);\quad
  {(0, {\sr c}) \over ({\sr c}, 0)} = (0, 1);\quad
  {(0, {\sr c}) \over (0, {\sr c})} = (1, 0);
   \eqno (46.)$$
and finally
$${({\sr c}, 0) \over (0, {\sr c})}
   =  \left( - {\gamma_2 \over \gamma_1}, {1 \over \gamma_1} \right),
   \eqno (47.)$$
because, when we make, in (43.),
$$\alpha_1 = 0,\quad \alpha_2 = 1,\quad
  \beta_1  = 1,\quad \beta_2  = 0,
   \eqno (48.)$$
we find
$$a_1 = - {\gamma_2 \over \gamma_1},\quad
  a_2 =   {1 \over \gamma_1};
   \eqno (49.)$$
so that although the ratio of the pure primary step-couple
$({\sr c}, 0)$ to the pure secondary step-couple $(0, {\sr c})$
can never be expressed as a {\it pure primary number-couple}, it
may be expressed as a {\it pure secondary number-couple}, namely
$\displaystyle \left( 0, {1 \over \gamma_1} \right)$,
if we choose~$0$, as in (36.), for the value of the secondary
constant~$\gamma_2$, but not otherwise: this choice
$\gamma_2 = 0$ is therefore required by simplicity.  And since by
the condition (45.), the primary constant~$\gamma_1$ must be
contra-positive, the simplest way of determining it is to make it
contra-positive one, $\gamma_1 = -1$, as announced in (36.).  We
have therefore justified that selection (36.) of the two
constants of multiplication; and find, with that selection,
$${({\sr c}, 0) \over (0, {\sr c})}
   =  (0, -1),
   \eqno (50.)$$
and generally, for the ratio of any one step-couple to any other,
the formula
$${({\sr b}_1, {\sr b}_2) \over ({\sr a}_1, {\sr a}_2)}
   =  {(\beta_1 {\sr c}, \beta_2 {\sr c})
         \over (\alpha_1 {\sr c}, \alpha_2 {\sr c})}
   =  \left(
         {\beta_1 \alpha_1 + \beta_2 \alpha_2
            \over \alpha_1^2 + \alpha_2^2}, \,
         {\beta_2 \alpha_1 - \beta_1 \alpha_2
            \over \alpha_1^2 + \alpha_2^2}
      \right).
   \eqno (51.)$$

\bigbreak

\centerline{\it
On the Addition, Subtraction, Multiplication, and Division, of
Number-Couples, as}

\nobreak\vskip 3pt

\centerline{\it
combined with each other.}

\nobreak\bigskip

6.
Proceeding to operations upon number-couples, considered in
combination with each other, it is easy now to see the
reasonableness of the following definitions, and even their
necessity, if we would preserve in the simplest way, the analogy
of the theory of couples to the theory of singles:
$$(b_1, b_2) + (a_1, a_2) = (b_1 + a_1, b_2 + a_2);
   \eqno (52.)$$
$$(b_1, b_2) - (a_1, a_2) = (b_1 - a_1, b_2 - a_2);
   \eqno (53.)$$
$$(b_1, b_2) (a_1, a_2)
    = (b_1, b_2) \times (a_1, a_2)
    = (b_1 a_1 - b_2 a_2, b_2 a_1 + b_1 a_2);
   \eqno (54.)$$
$${(b_1, b_2) \over (a_1, a_2)}
    = \left(
         {b_1 a_1 + b_2 a_2 \over a_1^2 + a_2^2}, \,
         {b_2 a_1 - b_1 a_2 \over a_1^2 + a_2^2}
      \right).
   \eqno (55.)$$
Were these definitions even altogether arbitrary, they would at
least not contradict each other, nor the earlier principles of
Algebra, and it would be possible to draw legitimate conclusions,
by rigorous mathematical reasoning, from premises thus
arbitrarily assumed: but the persons who have read with attention
the foregoing remarks of this theory, and have compared them with
the Preliminary Essay, will see that these definitions are really
{\it not arbitrarily chosen}, and that though others might have
been assumed, no others would be equally proper.

With these definitions, addition and subtraction of
number-couples are mutually inverse operations, and so are
multiplication and division; and we have the relations,
$$(b_1, b_2) + (a_1, a_2) = (a_1, a_2) + (b_1, b_2),
   \eqno (56.)$$
$$(b_1, b_2) \times (a_1, a_2) = (a_1, a_2) \times (b_1, b_2),
   \eqno (57.)$$
$$(b_1, b_2) \{ (a_1', a_2') + (a_1, a_2) \}
   =  (b_1, b_2) (a_1', a_2') + (b_1, b_2) (a_1, a_2):
   \eqno (58.)$$
we may, therefore, extend to number-couples all those results
respecting numbers, which have been deduced from principles
corresponding to these last relations.  For example,
$$\{ (b_1, b_2) + (a_1, a_2) \} \times
  \{ (b_1, b_2) + (a_1, a_2) \}
   =  (b_1, b_2) (b_1, b_2)
    + 2 (b_1, b_2) (a_1, a_2)
    + (a_1, a_2) (a_1, a_2),
   \eqno (59.)$$
in which
$$2 (b_1, b_2) (a_1, a_2)
   =  (2, 0) (b_1, b_2) (a_1, a_2)
   =  (b_1, b_2) (a_1, a_2) + (b_1, b_2) (a_1, a_2);
   \eqno (60.)$$
for, in general, we may {\it mix the signs of numbers with those
of number-couples}, if we consider every single number~$a$ as
equivalent to a pure primary number-couple,
$$a = (a, 0).
   \eqno (61.)$$
When the pure primary couple $(1, 0)$ is thus considered as
equivalent to the number~$1$, it may be called, for shortness,
the {\it primary unit\/}; and the pure secondary couple $(0, 1)$
may be called in like manner the {\it secondary unit}.

We may also agree to write, by analogy to notations already
explained,
$$\left. \eqalign{
(0, 0) + (a_1, a_2) = + (a_1, a_2),\cr
(0, 0) - (a_1, a_2) = - (a_1, a_2);\cr}
   \right\}
   \eqno (62.)$$
and then $+ (a_1, a_2)$ will be another symbol for the
number-couple $(a_1, a_2)$ itself, and $- (a_1, a_2)$ will be a
symbol for the {\it opposite number-couple\/} $(- a_1, - a_2)$.
The {\it reciprocal\/} of a number-couple $(a_1, a_2)$ is this
other number-couple,
$${1 \over (a_1, a_2)}
   =  {(1, 0) \over (a_1, a_2)}
   =  \left(
         {a_1 \over a_1^2 + a_2^2},
         {- a_2 \over a_1^2 + a_2^2}
      \right)
   =  {(a_1, - a_2) \over a_1^2 + a_2^2}.
   \eqno (63.)$$
It need scarcely be mentioned that the insertion of the sign of
coincidence~$=$ between any two number-couples implies that those
two couples coincide, number with number, primary with primary,
and secondary with secondary; so that {\it an equation between
number-couples is equivalent to a couple of equations between
numbers}.

\bigbreak

\centerline{\it
On the Powering of a Number-Couple by a Single Whole Number.}

\nobreak\bigskip

7.
Any number-couple $(a_1, a_2)$ may be used as a {\it base\/} to
generate a series of {\it powers}, with integer {\it exponents},
or {\it logarithms}, namely, the series
$$\ldots \,
   (a_1, a_2)^{-2},\quad
   (a_1, a_2)^{-1},\quad
   (a_1, a_2)^0,\quad
   (a_1, a_2)^1,\quad
   (a_1, a_2)^2,\ldots
   \eqno (64.)$$
in which the {\it first positive power\/} $(a_1, a_2)^1$ is the
base itself, and all the others are formed from it by repeated
multiplication or division by that base, according as they follow
or precede it in the series; thus,
$$(a_1, a_2)^0 = (1, 0),
   \eqno (65.)$$
and
$$\left. \multieqalign{
(a_1, a_2)^1    &= (a_1, a_2), &
(a_1, a_2)^{-1} &= {(1, 0) \over (a_1, a_2)}, \cr
(a_1, a_2)^2    &= (a_1, a_2) (a_1, a_2), &
(a_1, a_2)^{-2} &= {(1, 0) \over (a_1, a_2) (a_1, a_2)}, \cr
   &\hbox{\&c.} & &\hbox{\&c.} \cr}
   \right\}
   \eqno (66.)$$

To {\it power\/} the couple $(a_1, a_2)$ by any {\it positive\/}
whole number~$m$, is, therefore, to {\it multiply}, $m$ times
successively, the {\it primary unit}, or the couple $(1, 0)$, by
the proposed couple $(a_1, a_2)$; and to power $(a_1, a_2)$ by
any {\it contra-positive\/} whole number $-m$, is to
{\it divide\/} $(1, 0)$ by the same couple $(a_1, a_2)$, $m$
times successively: but to power by $0$ produces always $(1, 0)$.
Hence, generally, for any whole numbers $\mu$, $\nu$,
$$\left. \eqalign{
(a_1, a_2)^\mu (a_1, a_2)^\nu &= (a_1, a_2)^{\mu + \nu},\cr
( (a_1, a_2)^\mu )^\nu        &= (a_1, a_2)^{\mu \nu}.\cr}
   \right\}
   \eqno (67.)$$

\bigbreak

8.
In the theory of single numbers,
$$\eqalignno{
{(a + b)^m \over 1 \times 2 \times 3 \, \times \ldots m}
   &=    {a^m \over 1 \times 2 \times 3 \, \times \ldots m}
       + {a^{m-1} \over 1 \times 2 \times 3 \, \times \ldots (m-1)}
         {b^1 \over 1} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {a^{m-2} \over 1 \times 2 \times 3 \, \times \ldots (m-2)}
         {b^2 \over 1 \times 2}
       + \ldots \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {a^1 \over 1}
         {b^{m-1} \over 1 \times 2 \times 3 \, \times \ldots (m-1)}
       + {b^m \over 1 \times 2 \times 3 \, \times \ldots m};
   &(68.)\cr}$$
and similarly in the theory of number-couples,
$$\eqalignno{
{ \{ (a_1, a_2) + (b_1, b_2) \}^m
     \over 1 \times 2 \times 3 \, \times \ldots m}
   &=    {(a_1, a_2)^m \over 1 \times 2 \times 3 \, \times \ldots m}
       + {(a_1, a_2)^{m-1} \over 1 \times 2 \times 3 \, \times \ldots (m-1)}
         {(b_1, b_2)^1 \over 1} \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {(a_1, a_2)^{m-2} \over 1 \times 2 \times 3 \, \times \ldots (m-2)}
         {(b_1, b_2)^2 \over 1 \times 2}
       + \ldots \cr
   &\mathrel{\phantom{=}} \mathord{}
       + {(a_1, a_2)^1 \over 1}
         {(b_1, b_2)^{m-1} \over 1 \times 2 \times 3 \, \times \ldots (m-1)}
       + {(b_1, b_2)^m \over 1 \times 2 \times 3 \, \times \ldots m};
   &(69.)\cr}$$
$m$ being in both  these formul{\ae} a positive whole number, but
$a$~$b$~$a_1$~$a_2$~$b_1$~$b_2$ being any numbers whatever.  The
latter formula, which includes the former, may easily be proved
by considering the product of $m$ unequal factor sums,
$$(a_1, a_2) + (b_1^{(1)}, b_2^{(1)}),\quad
  (a_1, a_2) + (b_1^{(2)}, b_2^{(2)}),\quad\ldots\quad
  (a_1, a_2) + (b_1^{(m)}, b_2^{(m)});
   \eqno (70.)$$
for, in this product, when developed by the rules of
multiplication, the power $(a_1, a_2)^{m-n}$ is multiplied by the
sum of all the products of $n$ factor couples such as
$(b_1^{(1)}, b_2^{(1)})$
$(b_1^{(2)}, b_2^{(2)})$~$\ldots$
$(b_1^{(n)}, b_2^{(n)})$;
and the number of such products is the number of combinations of
$m$ things, taken $n$ by $n$, that is,
$${1 \times 2 \times 3 \times \ldots \times m
   \over 1 \times 2 \times 3 \times \ldots \, (m - n)
      \times 1 \times 2 \times 3 \times \ldots \, n},
   \eqno (71.)$$
while these products themselves become each $= (b_1, b_2)^n$,
when we return to the case of equal factors.

The formula (69.) enables us to determine separately the primary
and secondary numbers of the power or couple $(a_1, a_2)^m$, by
treating the base $(a_1, a_2)$ as the sum of a pure primary
couple $(a_1, 0)$ and a pure secondary $(0, a_2)$, and by
observing that the powering of these latter number-couples may be
performed by multiplying the powers of the numbers $a_1$~$a_2$ by
the powers of the primary and secondary units, $(1, 0)$ and
$(0, 1)$; for, whatever whole number~$i$ may be,
$$\left. \eqalign{
(a_1, 0)^i &= a_1^i (1,0)^i,\cr
(0, a_2)^i &= a_2^i (1,0)^i.\cr}
   \right\}
   \eqno (72.)$$
We have also the following expressions for the powers of these
two units,
$$\left. \eqalign{
(1, 0)^i &= (1, 0),\cr
(1, 0)^{4k-3} &= (0, 1),\cr
(1, 0)^{4k-2} &= (-1, 0),\cr
(1, 0)^{4k-1} &= (0, -1),\cr
(1, 0)^{4k}   &= (1, 0);\cr}
   \right\}
   \eqno (73.)$$
that is, the powers of the primary unit are all themselves equal
to that primary unit; but the first, second, third, and fourth
powers of the secondary unit are respectively
$$(0,1),\quad (-1,0),\quad (0,-1),\quad (1,0),$$
and the higher powers are formed by merely repeating this period.
In like manner we find that the equation
$$(a_1, a_2)^m = (b_1, b_2),
   \eqno (74.)$$
is equivalent to the two following,
$$\left. \eqalign{
b_1
   &= a_1^m
        - {m (m - 1)
            \over 1 \times 2}
            a_1^{m-2} a_2^2
       + {m (m - 1) (m - 2) (m - 3)
            \over 1 \times 2 \times 3 \times 4}
            a_1^{m-4} a_2^4
       - \quad\hbox{\&c.}\cr
b_2
   &= m a_1^{m-1} a_2
       - {m (m - 1) (m - 2)
            \over 1 \times 2 \times 3}
            a_1^{m-3} a_2^3
       + \quad\hbox{\&c.}\cr}
   \right\}
   \eqno (75.)$$
For example, the square and cube of a couple, that is, the second
and third positive powers of it, may be developed thus,
$$(a_1, a_2)^2
   =  \{ (a_1, 0) + (0, a_2) \}^2
   =  (a_1^2 - a_2^2, 2 a_1 a_2),
   \eqno (76.)$$
and
$$(a_1, a_2)^3
   =  \{ (a_1, 0) + (0, a_2) \}^3
   =  (a_1^3 - 3 a_1 a_2^2, 3 a_1^2 a_2 - a_2^3).
   \eqno (77.)$$

\bigbreak

9.
In general, if
$$(a_1, a_2) (a_1', a_2') = (a_1'', a_2''),
   \eqno (78.)$$
then, by the theorem or rule of multiplication (54.)
$$a_1'' = a_1 a_1' - a_2 a_2',\quad
  a_2'' = a_2 a_1' + a_1 a_2',
   \eqno (79.)$$
and therefore
$$a_1''^2 + a_2''^2 = (a_1^2 + a_2^2) (a_1'^2 + a_2'^2);
   \eqno (80.)$$
and in like manner it may be proved that
$$\left. \eqalign{
\hbox{if}\quad &
   (a_1, a_2) (a_1' a_2') (a_1'', a_2'')
      =  (a_1''', a_2'''),\cr
\hbox{then}\quad &
(a_1'''^2 + a_2'''^2)
   = (a_1^2 + a_2^2) (a_1'^2 + a_2'^2) (a_1''^2 + a_2''^2),\cr}
   \right\}
   \eqno (81.)$$
and so on, for any number~$m$ of factors.  Hence, in
particular, when all these $m$ factors are equal, so that the
product becomes a power, and the equation (74.) is satisfied, the
two numbers $b_1$~$b_2$ of the {\it power-couple\/} must be
connected with the two numbers $a_1$~$a_2$ of the
{\it base-couple\/} by the relation
$$b_1^2 + b_2^2 = (a_1^2 + a_2^2)^m.
   \eqno (82.)$$
For example, in the cases of the square and cube, this relation
holds good under the forms
$$(a_1^2 - a_2^2)^2 + (2 a_1 a_2)^2 = (a_1^2 + a_2^2)^2,
   \eqno (83.)$$
and
$$(a_1^3 - 3 a_1 a_2^2)^2 + (3 a_1^2 a_2 - a_2^3)^2
   =  (a_1^2 + a_2^2)^3.
   \eqno (84.)$$
The relation (82.) is true even for powers with contra-positive
exponents $-m$, that is,
$$b_1^2 + b_2^2 = (a_1^2 + a_2^2)^{-m}
   \quad\hbox{if}\quad
  (b_1, b_2) = (a_1, a_2)^{-m};
   \eqno (85.)$$
for in general
$$\left. \eqalign{
\hbox{if}\quad
(b_1, b_2)
   &= {(a_1, a_2) (a_1', a_2') (a_1'', a_2'') \ldots
         \over (c_1, c_2) (c_1', c_2') (c_1'', c_2'') \ldots} \cr
\hbox{then}\quad
(b_1^2 + b_2^2)
   &= {(a_1^2 + a_2^2) (a_1'^2 + a_2'^2) (a_1''^2 + a_2''^2) \ldots
         \over (c_1^2 + c_2^2)
               (c_1'^2 + c_2'^2)
               (c_1''^2 + c_2''^2) \ldots} \cr}
   \right\}
   \eqno (86.)$$

\bigbreak

\centerline{\it
On a particular Class of Exponential and Logarithmic
Function-Couples, connected}

\nobreak\vskip 3pt

\centerline{\it
with a particular Series of Integer Powers of Number-Couples.}

\nobreak\bigskip

10.
The theorem (69.) shows, that if we employ the symbols
${\sc f}_m(a_1, a_2)$ and ${\sc f}_m(b_1, b_2)$ to denote
concisely two number-couples, which depend in the following way
on the couples $(a_1, a_2)$ and $(b_1, b_2)$,
$$\eqalignno{
{\sc f}_m(a_1, a_2)
   &=    (1, 0)
       + {(a_1, a_2)^1 \over 1}
       + {(a_1, a_2)^2 \over 1 \times 2}
       + \ldots
       + {(a_1, a_2)^m \over 1 \times 2 \times 3 \times \ldots \, m},
   &(87.)\cr
{\sc f}_m(b_1, b_2)
   &=    (1, 0)
       + {(b_1, b_2)^1 \over 1}
       + {(b_1, b_2)^2 \over 1 \times 2}
       + \ldots
       + {(b_1, b_2)^m \over 1 \times 2 \times 3 \times \ldots \, m},
   &(88.)\cr}$$
and if we denote in like manner by the symbol
$${\sc f}_m ((a_1, a_2) + (b_1, b_2))
   =  {\sc f}_m (a_1 + b_1, a_2 + b_2)
   \eqno (89.)$$
the couple which depends in the same way on the sum
$(a_1, a_2) + (b_1, b_2)$, or on the couple
$(a_1 + a_2, b_1 + b_2)$, and develope by the rule (69.) the
powers of this latter sum, we shall have the relation
$$\eqalignno{
\{ {\sc f}_m(a_1, a_2) \times {\sc f}_m(b_1, b_2) \}
   - {\sc f}_m((a_1, a_2) + (b_1, b_2))
   \hskip-18em \cr
   &= {(a_1, a_2)^m \over
               1 \times 2 \times 3 \times \ldots \, m}
      \left\{
         {(b_1, b_2)^1 \over 1}
       + {(b_1, b_2)^2 \over 1 \times 2}
       + \ldots
       + {(b_1, b_2)^m \over
               1 \times 2 \times 3 \times \ldots \, m}
      \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {(a_1, a_2)^{m-1} \over
               1 \times 2 \times 3 \times \ldots \, (m - 1)}
      \left\{
         {(b_1, b_2)^2 \over 1 \times 2}
       + \ldots
       + {(b_1, b_2)^m \over
               1 \times 2 \times 3 \times \ldots \, m}
      \right\} \cr
   &\mathrel{\phantom{=}} \mathord{}
    + \ldots \cr
   &\mathrel{\phantom{=}} \mathord{}
    + {(a_1, a_2)^1 \over 1}
         {(b_1, b_2)^m \over
               1 \times 2 \times 3 \times \ldots \, m}.
   &(90.)\cr}$$
This expression may be farther developed, by the rule for the
multiplication of a sum, into the sum of several terms or
couples, $(c_1, c_2)$, of which the number is
$$1 + 2 + 3 + \ldots + m
   ={m (m + 1) \over 2},
   \eqno (91.)$$
and which are of the form
$$(c_1, c_2)
   =  {(a_1, a_2)^i \over
               1 \times 2 \times 3 \times \ldots \, i}
      \times
      {(b_1, b_2)^k \over
               1 \times 2 \times 3 \times \ldots \, k},
   \eqno (92.)$$
$i$ and $k$ being positive integers, such that
$$i \ngt m,\quad
  k \ngt m,\quad
   i + k > m;
   \eqno (93.)$$
and if we put for abridgement
$$\sqrt{a_1^2 + a_2^2} = \alpha,\quad
  \sqrt{b_1^2 + b_2^2} = \beta,
   \eqno (94.)$$
and
$$\gamma
   =  {\alpha^i \beta^k \over
               1 \times 2 \times 3 \times \ldots \, i
        \times 1 \times 2 \times 3 \times \ldots \, k},
   \eqno (95.)$$
we shall have, by principles lately explained,
$$\sqrt{ c_1^2 + c_2^2 } = \gamma,
   \eqno (96.)$$
and therefore
$$c_1 \ngt + \gamma,\quad
  c_1 \nlt - \gamma,\quad
  c_2 \ngt + \gamma,\quad
  c_2 \nlt - \gamma:
   \eqno (97.)$$
if then the entire sum (90.) of all these couples $(c_1, c_2)$ be
put under the form
$${\textstyle\sum} (c_1, c_2)
   = ( {\textstyle\sum} c_1, {\textstyle\sum} c_2 ),
   \eqno (98.)$$
the letter ${\textstyle\sum}$ being used as a mark of summation,
we shall have the corresponding limitations
$$\left. \multieqalign{
   {\textstyle\sum} c_1 &\ngt   {\textstyle\sum} \gamma, &
   {\textstyle\sum} c_1 &\nlt - {\textstyle\sum} \gamma, \cr
   {\textstyle\sum} c_2 &\ngt   {\textstyle\sum} \gamma, &
   {\textstyle\sum} c_2 &\nlt - {\textstyle\sum} \gamma, \cr}
   \right\}
   \eqno (99.)$$
${\textstyle\sum} \gamma$ being the positive sum of the
$\displaystyle {m (m + 1) \over 2}$
such terms as that marked (95.).  This latter sum depends on the
positive whole number~$m$, and on the positive numbers
$\alpha$,~$\beta$; but whatever these two latter numbers may be,
it is easy to show that by taking the former number sufficiently
great, we can make the positive sum ${\textstyle\sum} \gamma$
become smaller, that is nearer to $0$, than any positive
number~$\delta$ previously assigned, however small that
number~$\delta$ may be.  For if we use the symbols
${\sc f}_m(\alpha)$, ${\sc f}_m(\beta)$,
${\sc f}_m(\alpha + \beta)$, to denote positive numbers connected
with the positive numbers $\alpha$, $\beta$, $\alpha + \beta$, by
relations analogous to those marked (87.) and (88.), so that
$${\sc f}_m(\alpha)
   = 1 + {\alpha \over 1}
       + {\alpha^2 \over 1 \times 2}
       + \ldots
       + {\alpha^m \over 1 \times 2 \times 3 \times \ldots \, m},
   \eqno (100.)$$
it is easy to prove, by (68.), that the product
${\sc f}_m(\alpha) \times {\sc f}_m(\beta)$
exceeds the number ${\sc f}_m(\alpha + \beta)$ by
${\textstyle\sum} \gamma$, but falls short of the number
${\sc f}_{2m} (\alpha + \beta)$, that is of the following number
$${\sc f}_{2m} (\alpha + \beta)
   = 1 + {(\alpha + \beta)^1 \over 1}
       + {(\alpha + \beta)^2 \over 1 \times 2}
       + \ldots
       + {(\alpha + \beta)^{2m} \over
            1 \times 2 \times 3 \times \ldots \times 2m};
   \eqno (101.)$$
so that
$${\textstyle\sum} \gamma
   =  ({\sc f}_m (\alpha) \times {\sc f}_m (\beta))
       - {\sc f}_m (\alpha + \beta),
   \eqno (102.)$$
and
$${\textstyle\sum} \gamma
   <   {\sc f}_{2m} (\alpha + \beta)
       - {\sc f}_m (\alpha + \beta):
   \eqno (103.)$$
if then we choose a positive integer~$n$, so as to satisfy the
condition
$$n + 1 > 2 (\alpha + \beta),
   \quad\hbox{that is}\quad
   {\alpha + \beta \over n + 1} < {\textstyle {1 \over 2}},
   \eqno (104.)$$
and take $m > n$, we shall have
$${(\alpha + \beta)^m
         \over 1 \times 2 \times 3 \times \ldots \, m}
   <  {1 \over 2^{m - n}}
      {(\alpha + \beta)^n
         \over 1 \times 2 \times 3 \times \ldots \, n},
   \quad\hbox{and therefore}\quad
   < \delta,
   \eqno (105.)$$
however small the positive number~$\delta$ may be, and however
large $\alpha + \beta$ may be, if we take $m$ large enough; but
also
$${\sc f}_{2m} (\alpha + \beta) - {\sc f}_m (\alpha + \beta)
   =  {(\alpha + \beta)^m \eta
         \over 1 \times 2 \times 3 \times \ldots \, m}
   \quad\hbox{and therefore}\quad
      < \delta \times \eta,
   \eqno (106.)$$
in which
$$\eta = {\alpha + \beta \over m + 1}
       + {(\alpha + \beta)^2 \over (m + 1) (m + 2)}
       + \ldots
       + {(\alpha + \beta)^m
         \over (m + 1) (m + 2) \times \ldots \, (2m)},
   \eqno (107.)$$
and, therefore,
$$\eta < 1,
   \eqno (108.)$$
because
$${\alpha + \beta \over m + 1}
      <  {1 \over 2},\quad
  {(\alpha + \beta)^2 \over (m + 1) (m + 2)}
      <  {1 \over 2^2},\quad\ldots\quad
  {(\alpha + \beta)^m \over (m + 1) (m + 2) \times \ldots (2m)};
      <  {1 \over 2^m};
   \eqno (109.)$$
therefore, combining the inequalities (103.) (106.) (108.), we
find finally
$${\textstyle\sum} \gamma < \delta.
   \eqno (110.)$$
And hence, by (99.), the two sums
${\textstyle\sum} c_1$,
${\textstyle\sum} c_2$,
may both be made to approach as near as we desire to $0$, by
taking $m$ sufficiently large; so that, in the notation of limits
already employed,
$$\underline{\rm L} {\textstyle\sum} \gamma = 0,\quad
  \underline{\rm L} {\textstyle\sum} c_1    = 0,\quad
  \underline{\rm L} {\textstyle\sum} c_2    = 0,
   \eqno (111.)$$
and, therefore,
$$\underline{\rm L}
   \{ {\sc f}_m (\alpha) {\sc f}_m (\beta)
       - {\sc f}_m (\alpha + \beta) \}
   =  0,
   \eqno (112.)$$
$$\underline{\rm L}
   \{ {\sc f}_m (a_1, a_2) {\sc f}_m (b_1, b_2)
         - {\sc f}_m ((a_1, a_2) + (b_1, b_2)) \}
   =  (0, 0).
   \eqno (113.)$$
In the foregoing investigation, $\alpha$ and $\beta$ denoted
positive numbers; but the theorem~(113.) shows that the
formula~(112.) holds good, whatever numbers may be denoted by
$\alpha$ and $\beta$, if we still interpret the symbol
${\sc f}_m(\alpha)$ by the rule~(100.).


\bigbreak

11.
If $\alpha$ still retain the signification (94.), it results,
from the foregoing reasonings, that the primary and secondary
numbers of the couple
$${\sc f}_{m+m'} (a_1, a_2) - {\sc f}_m (a_1, a_2)
   \eqno (114.)$$
are each
$$\ngt {\sc f}_{m+m'} (\alpha) - {\sc f}_m (\alpha),
   \quad\hbox{and}\quad
  \nlt {\sc f}_m (\alpha) - {\sc f}_{m+m'} (\alpha);
   \eqno (115.)$$
and, therefore, may each be made nearer to $0$ (on the positive
or on the contra-positive side) than any proposed positive
number~$\delta$ by choosing $m$ large enough, however large $m'$
and $\alpha$ may be, and however small $\delta$ may be: because
in the expression
$$\eqalignno{{\sc f}_{m+m'} (\alpha) - {\sc f}_m(\alpha)
   \hskip-84pt \cr
   &= {\alpha^m \over 1 \times 2 \times 3 \times \ldots \, m}
      \left\{
         {\alpha \over m + 1}
       + {\alpha^2 \over (m + 1) (m + 2)}
       + \ldots
       + {\alpha^{m'} \over (m + 1) \, \ldots \, (m + m')}
      \right\}
   &(116.)\cr}$$
the positive factor
$\displaystyle
   {\alpha^m \over 1 \times 2 \times 3 \times \ldots \, m}$
may be made $< \delta$, that is, as near as we please to $0$, and
also the other factor, as being
$\displaystyle
   < {1 \over n} + {1 \over n^2} + \ldots + {1 \over n^{m'}}$,
and therefore
$\displaystyle < {1 \over n - 1}$,
if $m + 1 > n \alpha$.
Pursuing this train of reasoning, we find that as $m$ becomes
greater and greater without end, the couple
${\sc f}_m (a_1, a_2)$ tends to a determinate {\it limit-couple},
which depends on the couple $(a_1, a_2)$, and may be denoted by
the symbol ${\sc f}_\infty (a_1, a_2)$, or simply
${\sc f}(a_1, a_2)$,
$${\sc f}(a_1, a_2) = {\sc f}_\infty (a_1, a_2)
   = \underline{\rm L} {\sc f}_m (a_1, a_2);
   \eqno (117.)$$
and similarly, that for any determinate number~$\alpha$, whether
positive or not, the number ${\sc f}_m(\alpha)$ tends to a
determinate {\it limit-number}, which depends on the
number~$\alpha$, and may be denoted thus,
$${\sc f}(\alpha) = {\sc f}_\infty (\alpha)
   = \underline{\rm L} {\sc f}_m (\alpha).
   \eqno (118.)$$
It is easy also to prove, by (112.), that this {\it function}, or
{\it dependent number}, ${\sc f}(\alpha)$, must always satisfy
the conditions
$${\sc f}(\alpha) \times {\sc f}(\beta)
   = {\sc f}(\alpha + \beta),
   \eqno (119.)$$
and that it increases constantly and continuously from positive
states indefinitely near to $0$ to positive states indefinitely
far from $0$, which $\alpha$ increases or advances constantly,
and continuously, and indefinitely in the progression from
contra-positive to positive; so that, for every positive
number~$\beta$, there is some determined number~$\alpha$ which
satisfies the condition
$$\beta = {\sc f}(\alpha),
   \eqno (120.)$$
and which may be thus denoted,
$$\alpha = {\sc f}^{-1}(\beta).
   \eqno (121.)$$
It may also be easily proved that we have always the relations,
$${\sc f}(\alpha) = \subzero{e}^\alpha,\quad
  {\sc f}^{-1}(\beta) = \log_e \mathord{} \mathbin{.} \beta,
   \eqno (122.)$$
if we put, for abridgement,
$${\sc f}(1) = e,
   \eqno (123.)$$
and employ the notation of powers and logarithms explained in the
Preliminary Essay.  A power $\subzero{b}^\alpha$, when considered
as depending on its exponent, is called an {\it exponential
function\/} thereof; its most general and essential properties
are those expressed by the formul{\ae},
$$\subzero{b}^\alpha \times \subzero{b}^\beta
   = \subzero{b}^{\alpha + \beta},\quad
  \subzero{b}^1 = b,
   \eqno (124.)$$
of which the first is independent of the base~$b$, while the
second specifies that base; and since, by (113.), the
function-couple ${\sc f}(a_1, a_2)$ satisfies the analogous
condition,
$${\sc f}(a_1, a_2) \times {\sc f}(b_1, b_2)
   =  {\sc f}((a_1, a_2) + (b_1, b_2))
   =  {\sc f}(a_1 + b_1, a_2 + b_2),
   \eqno (125.)$$
(whatever numbers $a_1$~$a_2$~$b_1$~$b_2$ may be,) we may say by
analogy that this function-couple ${\sc f}(a_1, a_2)$ is an
{\it exponential function-couple}, and that its
{\it base-couple\/} is
$${\sc f}(1, 0) = (e, 0):
   \eqno (126.)$$
and because the exponent~$\alpha$ of a
power~$\subzero{b}^\alpha$, when considered as depending on that
power, is called a {\it logarithmic function\/} thereof, we may
say by analogy that the couple $(a_1, a_2)$ is a {\it logarithmic
function}, or {\it function-couple}, of the couple
${\sc f}(a_1, a_2)$, and may denote it thus,
$$(a_1, a_2) = {\sc f}^{-1} (b_1, b_2),
   \quad\hbox{if}\quad
  (b_1, b_2) = {\sc f}(a_1, a_2).
   \eqno (127.)$$
In general, if we can discover any law of dependence of one
couple $\Phi(a_1, a_2)$, upon another $(a_1, a_2)$, such that for
all values of the numbers $a_1$~$a_2$~$b_1$~$b_2$ the condition
$$\Phi(a_1, a_2) \Phi(b_1, b_2)
   =  \Phi(a_1 + a_2, b_1 + b_2)
   \eqno (128.)$$
is satisfied, then, whether this function-couple $\Phi(a_1, a_2)$
be or be not coincident with the particular function-couple
${\sc f}(a_1, a_2)$, we may call it (by the same analogy of
definition) an {\it exponential function-couple}, calling the
particular couple $\Phi(1, 0)$ its {\it base}, or
{\it base-couple\/}; and may call the couple $(a_1, a_2)$, when
considered as depending inversely on $\Phi(a_1, a_2)$, a
{\it logarithmic function}, or {\it function-couple}, which we
may thus denote,
$$(a_1, a_2) = \Phi^{-1} (b_1, b_2),
   \quad\hbox{if}\quad
  (b_1, b_2) = \Phi(a_1, a_2).
   \eqno (129.)$$

\bigbreak

12.
We have shown that the particular exponential function-couple
$(b_1, b_2) = {\sc f}(a_1, a_2)$ is always possible and
determinate, whatever the determinate couple $(a_1, a_2)$ may be;
let us now consider whether, inversely, the particular
logarithmic function-couple $(a_1, a_2) = {\sc f}^{-1}(b_1, b_2)$
is always possible and determinate, for every determined couple
$(b_1, b_2)$.  By the exponential properties of the
function~${\sc f}$, we have
$$\eqalignno{
(b_1, b_2) = {\sc f}(a_1, a_2)
   &= {\sc f}(a_1, 0) {\sc f}(0, a_2)
    = {\sc f}(a_1) {\sc f}(0 , a_2) \cr
   &= (\subzero{e}^{a_1} \cos a_2, \subzero{e}^{a_1} \sin a_2),
      &(130.)\cr}$$
if we define the functions $\cos \alpha$ and $\sin \alpha$, or
more fully the {\it cosine\/} and {\it sine\/} of any
number~$\alpha$, to be the primary and secondary numbers of the
couple ${\sc f}(0, \alpha)$, or the numbers which satisfy the
{\it couple-equation},
$${\sc f}(0, \alpha) = (\cos \alpha, \sin \alpha).
   \eqno (131.)$$
From this definition, or from these two others which it includes,
namely from the following expressions of the functions
{\it cosine\/} and {\it sine\/} as limits of the sums of series,
which are already familiar to mathematicians,
$$\left. \eqalign{
\cos \alpha
   &= 1
       - {\alpha^2 \over 1 \times 2}
       + {\alpha^4 \over 1 \times 2 \times 3 \times 4}
       - \quad\hbox{\&c.} \cr
\sin \alpha
   &= \alpha
       - {\alpha^3 \over 1 \times 2 \times 3}
       + {\alpha^5 \over 1 \times 2 \times 3 \times 4 \times 5}
       - \quad\hbox{\&c.} \cr}
   \right\}
   \eqno (132.)$$
it is possible to deduce all the other known properties of these
two functions; and especially that they are {\it periodical
functions}, in such a manner that while the variable
number~$\alpha$ increases constantly and continuously from $0$ to
a certain constant positive number
$\displaystyle {\pi \over 2}$, ($\pi$ being a certain number
between $3$ and $4$,) the function $\sin \alpha$ increases with
it (constantly and continuously) from $0$ to $1$, but
$\cos \alpha$ decreases (constantly and continuously) from $1$
to $0$; while $\alpha$ continues to increase from
$\displaystyle {\pi \over 2}$ to $\pi$, $\sin \alpha$ decreases
from $1$ to $0$, and $\cos \alpha$ from $0$ to $-1$; while
$\alpha$ increases from $\pi$ to
$\displaystyle {3\pi \over 2}$, $\sin\alpha$ decreases from $0$
to $-1$, but $\cos \alpha$ increases from $-1$ to $0$; while
$\alpha$ still increases from
$\displaystyle {3\pi \over 2}$ to $2\pi$, $\sin\alpha$ increases
from $-1$ to $0$, and $\cos \alpha$ from $0$ to $1$, the sum of
the squares of the cosine and sine remaining always $= 1$; and
that then the same changes recur in the same order, having also
occurred before for contra-positive values of $\alpha$, according
to this {\it law of periodicity}, that
$$\cos (\alpha \pm 2i \pi) = \cos \alpha,\quad
  \sin (\alpha \pm 2i \pi) = \sin \alpha,
   \eqno (133.)$$
$i$ denoting here (as elsewhere in the present paper) any
positive whole number.  But because the proof of these well known
properties may be deduced from the equations (132.), without any
special reference to the theory of couples, it is not necessary,
and it might not be proper, to dwell upon it here.

It is, however, important to observe here, that by these
properties we can always find (or conceive found) an indefinite
variety of numbers~$\alpha$, differing from each other by
multiples of the constant number $2\pi$, and yet each having its
cosine equal to any one proposed number~$\beta_1$, and its sine
equal to any other proposed number~$\beta_2$, provided that the
sum of the squares of these two proposed numbers
$\beta_1$,~$\beta_2$, is $= 1$; and reciprocally, that if two
different numbers~$\alpha$ both satisfy the conditions
$$\cos \alpha = \beta_1,\quad \sin \alpha = \beta_2,
   \eqno (134.)$$
$\beta_1$ and $\beta_2$ being two given numbers, such that
$\beta_1^2 + \beta_2^2 = 1$,
then the difference of these two numbers~$\alpha$ is necessarily
a multiple of $2\pi$.  Among all these numbers~$\alpha$, there
will always be one which will satisfy these other conditions
$$\alpha > - \pi,\quad \alpha \ngt \pi,
   \eqno (135.)$$
and this particular number~$\alpha$ may be called the
{\it principal solution\/} of the equations (134.), because it is
always nearer to $0$ than any other number~$\alpha$ which
satisfies the same equations, except in the particular case when
$\beta_1 = -1$, $\beta_2 = 0$; and because, in this particular
case, though the two numbers $\pi$ and $-\pi$ are equally near
to $0$, and both satisfy the equations (134.), yet still the
principal solution~$\pi$, assigned by the conditions (135.) is
simpler than the other solution $-\pi$, which is rejected by
those last conditions.  It therefore always possible to find not
only one, but infinitely many number-couples $(a_1, a_2)$,
differing from each other by multiples of the constant couple
$(0, 2\pi)$, but satisfying each the equation (130.), and
therefore each entitled to be represented by, or included in the
meaning of, the general symbol ${\sc f}^{-1} (b_1, b_2)$,
whatever proposed effective couple $(b_1, b_2)$ may be.  For we
have only to satisfy, by (130.), the two separate equations
$$\subzero{e}^{a_1} \cos a_2 = b_1,\quad
  \subzero{e}^{a_1} \sin a_2 = b_2;
   \eqno (136.)$$
which are equivalent to the three following,
$$\subzero{e}^{a_1} = \sqrt{ b_1^2 + b_2^2 },
   \eqno (137.)$$
and
$$\cos a_2 = {b_1 \over \sqrt{ b_1^2 + b_2^2 }},\quad
  \sin a_2 = {b_2 \over \sqrt{ b_1^2 + b_2^2 }};
   \eqno (138.)$$
and if $\alpha$ be the {\it principal solution\/} of these two
last equations, we shall have as their most general solution
$$a_2 = \alpha + 2 \omega \pi,
   \eqno (139.)$$
while the formula (137.) gives
$$a_1 = \log_e \mathord{} \mathbin{.} \sqrt{ b_1^2 + b_2^2 }:
   \eqno (140.)$$
the couple $(a_1, a_2)$ admits therefore of all the following
values, consistently with the conditions (130.) or (136.),
$$(a_1, a_2) = {\sc f}^{-1} (b_1, b_2)
   =  ( \log_e \mathord{} \mathbin{.} \sqrt{ b_1^2 + b_2^2 },
            \alpha + 2 \omega \pi ),
   \eqno (141.)$$
in which $\omega$ is any whole number, and $\alpha$ is a number
$> - \pi$, but not $> \pi$, which has its cosine and sine
respectively equal to the proposed numbers $b_1$,~$b_2$, divided
each by the square-root of the sum of their squares.  To specify
any one value of $(a_1, a_2)$, or ${\sc f}^{-1}(b_1, b_2)$,
corresponding to any one particular whole number~$\omega$, we may
use the symbol $\putsub{\omega}{{\sc f}}^{-1}(b_1, b_2)$; and
then the symbol $\subzero{{\sc f}}^{-1}(b_1, b_2)$ will denote
what may be called the {\it principal value\/} of the inverse or
logarithmic function-couple ${\sc f}^{-1}(b_1, b_2)$, because it
corresponds to the principal value of the number~$a_2$, as
determined by the conditions (138.).

\bigbreak

\centerline{\it
On the Powering of any Number-Couple by any Single Number or
Number-Couple.}

\nobreak\bigskip

13.
Resuming now the problem of powering a number-couple by a number,
we may employ this property of the exponential
function~${\sc f}$,
$$({\sc f}(a_1, a_2))^\mu = {\sc f}(\mu a_1, \mu a_2),
   \eqno (142.)$$
$\mu$ being any whole number whether positive or contra-positive
or null; which easily follows from (125.), and gives this
expression for the $\mu$'th power, or {\it power-couple}, of any
effective number-couple,
$$(b_1, b_2)^\mu = {\sc f}(\mu {\sc f}^{-1} (b_1, b_2)).
   \eqno (143.)$$
Reciprocally if $(a_1, a_2)$ be an $m$th root, or
{\it root-couple}, of a proposed couple $(b_1, b_2)$, so that the
equation (74.) is satisfied, then
$$(a_1, a_2) = (b_1, b_2)^{1 \over m}
   =  {\sc f} \left( {1 \over m} {\sc f}^{-1} (b_1, b_2) \right).
   \eqno (144.)$$
This last expression admits of many values, when the positive
whole number~$m$ is $> 1$, on account of the indeterminateness of
the inverse or logarithmic function~${\sc f}^{-1}$; and to
specify any one of these values of the root-couple, corresponding
to any one value $\putsub{\omega}{{\sc f}}^{-1}$ of that inverse
function, which value of the root we may call {\it the
$\omega$'th value\/} of that root, we may employ the notation
$$\putsub{\omega}{(b_1, b_2)}^{1 \over m}
   =  {\sc f} \left( {1 \over m}
      \putsub{\omega}{{\sc f}}^{-1} (b_1, b_2) \right);
   \eqno (145.)$$
we may also call the particular value
$$\subzero{(b_1, b_2)}^{1 \over m}
   =  {\sc f} \left( {1 \over m}
      \subzero{{\sc f}}^{-1} (b_1, b_2) \right),
   \eqno (146.)$$
the {\it principal value\/} of the root-couple, or the
{\it principal $m$'th root of the couple\/}~$(b_1, b_2)$.  In
this notation,
$$\putsub{\omega}{(1, 0)}^{1 \over m}
   =  {\sc f} \left( 0, {2 \omega \pi \over m} \right),
   \eqno (147.)$$
$$\putsub{\omega}{(b_1, b_2)}^{1 \over m}
   =  \subzero{(b_1, b_2)}^{1 \over m}
      \putsub{\omega}{(1, 0)}^{1 \over m};
   \eqno (148.)$$
so that generally, {\it the $\omega$'th value of the $m$'th root
of any number-couple is equal to the principal value of that root
multiplied by the $\omega$'th value of the $m$'th root of the
primary unit\/}~$(1, 0)$.  The $m$th root of any couple has
therefore $m$ {\it distinct values}, and no more, because the
$m$th root of the primary unit $(1, 0)$ has $m$ distinct values,
and no more, since it may be thus expressed, by (147.) and
(131.),
$$\putsub{\omega}{(1, 0)}^{1 \over m}
   =  \left(
         \mathop{\rm cos.} {2 \omega \pi \over m},
         \mathop{\rm sin.} {2 \omega \pi \over m}
      \right),
   \eqno (149.)$$
so that, by the law of periodicity (133.), for any different
whole number~$\omega'$,
$$\putsub{\omega'}{(1, 0)}^{1 \over m}
   =  \putsub{\omega}{(1, 0)}^{1 \over m},
   \eqno (150.)$$
and therefore generally,
$$\putsub{\omega'}{(b_1, b_2)}^{1 \over m}
   =  \putsub{\omega}{(b_1, b_2)}^{1 \over m},
   \eqno (151.)$$
if
$$\omega' = \omega \pm i m,
   \eqno (152.)$$
but not otherwise.  For example, the cube-root of the primary
unit $(1, 0)$ has three distinct values, and no more, namely
$$\putsub{0}{(1, 0)}^{1 \over 3}
   = (1, 0);\quad
  \putsub{1}{(1, 0)}^{1 \over 3}
   = \left( - {1 \over 2}, {\surd 3 \over 2} \right);\quad
  \putsub{2}{(1, 0)}^{1 \over 3}
   = \left( - {1 \over 2}, - {\surd 3 \over 2} \right);
   \eqno (153.)$$
so that each of these three couples, but no other, has its cube
$= (1, 0)$.  Again the couple $(-1, 0)$ has two distinct
square-roots, and no more, namely
$$\putsub{0}{(-1, 0)}^{1 \over 2} = (0, 1);\quad
  \putsub{1}{(-1, 0)}^{1 \over 2} = (0, -1).
   \eqno (154.)$$
In general we may agree to denote the {\it principal
square-root\/} of a couple $(b_1, b_2)$ by the symbol
$$\sqrt{ (b_1, b_2) } = \subzero{(b_1, b_2)}^{1 \over 2};
   \eqno (155.)$$
and then we shall have the particular equation
$$\sqrt{ (-1, 0) } = (0,1);
   \eqno (156.)$$
which may, by the principle (61.), be concisely denoted as
follows,
$$\sqrt{-1} = (0,1).
   \eqno (157.)$$
In the {\sc theory of single numbers}, the symbol $\sqrt{-1}$ is
{\it absurd}, and denotes an {\sc impossible extraction}, or a
merely {\sc imaginary number}; but in the {\sc theory of
couples}, the same symbol $\sqrt{-1}$ is {\it significant}, and
denotes a {\sc possible extraction}, or a {\sc real couple},
namely (as we have just now seen) the {\it principal square-root
of the couple\/} $(-1, 0)$.  In the latter theory, therefore,
though not in the former, this sign $\sqrt{-1}$ may properly be
employed; and we may write, if we choose, for any couple
$(a_1, a_2)$ whatever,
$$(a_1, a_2)= a_1 + a_2 \sqrt{-1},
   \eqno (158.)$$
interpreting the symbols $a_1$ and $a_2$, in the expression
$a_1 + a_2 \sqrt{-1}$, as denoting the pure primary couples
$(a_1, 0)$ $(a_2, 0)$, according to the law of mixture (61.) of
numbers with number-couples, and interpreting the symbol
$\sqrt{-1}$, in the same expression, as denoting the secondary
unit or pure secondary couple $(0, 1)$, according to the
formula~(157.).  However, the notation $(a_1, a_2)$ appears to be
sufficiently simple.

\bigbreak

14.
In like manner, if we write, by analogy to the notation of
fractional powers of numbers,
$$(c_1, c_2) = (b_1, b_2)^{\nu \over \mu},
   \eqno (159.)$$
whenever the two couples $(b_1, b_2)$ and $(c_1, c_2)$ are both
related as integer powers to one common base couple $(a_1, a_2)$
as follows,
$$(b_1, b_2) = (a_1, a_2)^\mu,\quad
  (c_1, c_2) = (a_1, a_2)^\nu,
   \eqno (160.)$$
($\mu$ and $\nu$ being any two whole numbers, of which $\mu$ at
least is different from $0$,) we can easily prove that this
{\it fractional power-couple\/} $(c_1, c_2)$, or this result of
powering the couple $(b_1, b_2)$ by the fractional number
$\displaystyle {\nu \over \mu}$,
has in general many values, which are all expressed by the
formula
$$(c_1, c_2) = (b_1, b_2)^{\nu \over \mu}
   =  {\sc f} \left( {\nu \over \mu}
         {\sc f}^{-1} (b_1, b_2) \right),
   \eqno (161.)$$
and of which any one may be distinguished from the others by the
notation
$$\putsub{\omega}{(b_1, b_2)}^{\nu \over \mu}
   =  {\sc f} \left(
            {\nu \over \mu}
            \putsub{\omega}{{\sc f}}^{-1} (b_1, b_2)
         \right).
   \eqno (162.)$$
We may call the couple thus denoted {\it the $\omega$'th value of
the fractional power}, and in particular we may call
$$\subzero{(b_1, b_2)}^{\nu \over \mu}
   =  {\sc f}  \left(
            {\nu \over \mu}
            \subzero{{\sc f}}^{-1} (b_1, b_2)
         \right)
   \eqno (163.)$$
the {\it principal value}.  The $\omega$'th value may be formed
from the principal value, by multiplying it by the $\omega$'th
value of the corresponding fractional power of the primary unit,
that is, by the following couple,
$$\putsub{\omega}{(1, 0)}^{\nu \over \mu}
   =     \left(
            \cos {2 \omega \nu \pi \over \mu},
            \sin {2 \omega \nu \pi \over \mu},
         \right);
   \eqno (164.)$$
and therefore the number of distinct values of any fractional
power of a couple, is equal to the number~$m$ of units which
remain in the denominator, when the fraction
$\displaystyle {\nu \over \mu}$
has been reduced to its simplest possible expresssion, by the
rejection of common factors.

\bigbreak

15.
Thus, the {\it powering of any couple~$(b_1, b_2)$ by any
commensurable number\/}~$x$ may be effected by the formula,
$$(b_1, b_2)^x = {\sc f}(x {\sc f}^{-1}(b_1, b_2));
   \eqno (165.)$$
or by these more specific expressions,
$$\eqalignno{
\putsub{\omega}{(b_1, b_2)}^x
   &= {\sc f}(x \putsub{\omega}{{\sc f}}^{-1}(b_1, b_2)) \cr
   &= \subzero{(b_1, b_2)}^x \putsub{\omega}{(1, 0)}^x,
   &(166.)\cr}$$
in which
$$\putsub{\omega}{(1, 0)}^x
   =  ( \cos \overline{2 \omega x \pi},
        \sin \overline{2 \omega x \pi} ):
   \eqno (167.)$$
and it is natural to extend the same formul{\ae} by definition,
for reasons of analogy and continuity, even to the case when the
exponent or number~$x$ is {\it incommensurable}, in which latter
case {\it the variety of values of the power is infinite, though
no confusion can arise, if each be distinguished from the others
by its specific ordinal number}, or {\it determining
integer\/}~$\omega$.

And since the spirit of the present theory leads us to extend all
operations with single numbers to operations with number-couples,
we shall further define (being authorised by this analogy to do
so) that {\it the powering of any one number-couple $(b_1, b_2)$
by any other number-couple\/} $(x_1, x_2)$ is the calculation of
a third number-couple $(c_1, c_2)$ such that
$$(c_1, c_2) = (b_1, b_2)^{(x_1, x_2)}
   =  {\sc f}((x_1, x_2) \times {\sc f}^{-1} (b_1, b_2));
   \eqno (168.)$$
or more specifically of any one of the infinitely many couples
corresponding to the infinite variety of {\it specific
ordinals\/} or {\it determining integers~$\omega$}, according to
this formula,
$$\eqalignno{
\putsub{\omega}{(b_1, b_2)}^{(x_1, x_2)}
   &= {\sc f}((x_1, x_2) \times
         \putsub{\omega}{{\sc f}}^{-1} (b_1, b_2)) \cr
   &= \subzero{(b_1, b_2)}^{(x_1, x_2)}
         \putsub{\omega}{(1, 0)}^{(x_1, x_2)},
      &(169.)\cr}$$
in which the factor $\subzero{(b_1, b_2)}^{(x_1, x_2)}$ may be
called the {\it principal value\/} of the general power-couple,
and in which the other factor may be calculated by the following
expression,
$$\eqalignno{
\putsub{\omega}{(1, 0)}^{(x_1, x_2)}
   &= {\sc f}((x_1, x_2) \times (0, 2 \omega \pi)) \cr
   &= {\sc f}( - 2 \omega \pi x_2, 2 \omega \pi x_1 ) \cr
   &= \subzero{e}^{-2 \omega \pi x_2}
         (\cos 2 \omega \pi x_1, \sin 2 \omega \pi x_1 ).
      &(170.)\cr}$$
For example,
$$\subzero{(1, 0)}^{(x_1, x_2)}
   =  (1, 0),
   \eqno (171.)$$
and
$$\subzero{(e, 0)}^{(x_1, x_2)}
   =  {\sc f}(x_1, x_2);
   \eqno (172.)$$
also
$$\putsub{\omega}{(e, 0)}^{(x_1, x_2)}
   = {\sc f}((x_1, x_2) \times (1, 2 \omega \pi)).
   \eqno (173.)$$

\bigbreak

\centerline{\it
On Exponential and Logarithmic Function-Couples in general.}

\nobreak\bigskip

16.
It is easy now to discover this general expression for an
exponential function-couple:
$$\Phi(x_1, x_2)
   = {\sc f}((x_1, x_2) \times (a_1, a_2));
   \eqno (174.)$$
in which $(a_1, a_2)$ is any constant couple, independent of
$(x_1, x_2)$.  This {\it general exponential function\/}~$\Phi$
includes the particular function~${\sc f}$, and satisfies (as it
ought) the condition of the form (128.),
$$\Phi(x_1, x_2) \Phi(y_1, y_2)
   = \Phi(x_1 + y_1, x_2 + y_2);
   \eqno (175.)$$
its {\it base}, or {\it base-couple}, which may be denoted for
conciseness by $(b_1, b_2)$, is, by the 11th article, the couple
$$(b_1, b_2) = \Phi(1, 0) = {\sc f}(a_1, a_2);
   \eqno (176.)$$
and if we determine that integer number~$\omega$ which satisfies
the conditions
$$a_2 - 2 \omega \pi > - \pi,\quad
  a_2 - 2 \omega \pi \ngt \pi,
   \eqno (177.)$$
we shall have the general transformation
$$\Phi(x_1, x_2) = \putsub{\omega}{(b_1, b_2)}^{(x_1, x_2)}.
   \eqno (178.)$$
And the general {\it inverse exponential\/} or {\it logarithmic
function-couple}, which may, by (129.), be thus denoted,
$$(x_1, x_2) = \Phi^{-1}(y_1, y_2),
   \quad\hbox{if}\quad
  (y_1, y_2) = \Phi(x_1, x_2),
   \eqno (179.)$$
may also, by (174.) and (176.), be thus expressed:
$$\Phi^{-1}(y_1, y_2)
   =  {{\sc f}^{-1}(y_1, y_2) \over {\sc f}^{-1}(b_1, b_2)};
   \eqno (180.)$$
it involves, therefore, {\it two arbitrary integer numbers}, when
only the couple $(y_1, y_2)$ and the base $(b_1, b_2)$ are given,
and it may be thus more fully written,
$$\mathord{\mathop{\Phi}\limits^{\omega'}_{\omega}}{}^{-1} (y_1, y_2)
   =  \mathord{\mathop{\rm log}\limits^{\omega'}_{\omega}}{}_{(b_1, b_2)}
         \mathord{} \mathbin{.} (y_1, y_2)
   =  {\putsub{\omega'} {\sc f}^{-1} (y_1, y_2)
          \over \putsub{\omega} {\sc f}^{-1} (b_1, b_2)}.
   \eqno (181.)$$
For example, the general expression for the logarithms of the
primary unit $(1,0)$ to the base $(e,0)$, is
$$\mathord{\mathop{\rm log}\limits^{\omega'}_{\omega}}{}_{(e, 0)}
         \mathord{} \mathbin{.} (1, 0)
   =  {(0, 2 \omega' \pi) \over (1 , 2 \omega \pi)}
   =  {(2 \omega' \pi, 0) \over (2 \omega \pi, -1)},
   \eqno (182.)$$
or, if we choose to introduce the symbol $\sqrt{-1}$, {\it as
explained in the\/ {\rm 13}th article}, that is, as denoting the
couple $(0, 1)$ according to the law of mixture of numbers with
number-couples, then
$$\mathord{\mathop{\rm log}\limits^{\omega'}_{\omega}}{}_e
         \mathord{} \mathbin{.} 1
   =  {2 \omega' \pi \sqrt{-1} \over 1 + 2 \omega \pi \sqrt{-1}}
   =  {2 \omega' \pi \over 2 \omega \pi - \sqrt{-1}}.
   \eqno (183.)$$
In general,
$$\mathord{\mathop{\rm log}\limits^{\omega'}_{\omega}}{}_{(b_1, b_2)}
         \mathord{} \mathbin{.} (y_1, y_2)
   =  {\subzero{{\sc f}}^{-1}(y_1, y_2) + (0, 2 \omega' \pi)
          \over \subzero{{\sc f}}^{-1}(b_1, b_2) + (0, 2 \omega \pi)}.
   \eqno (184.)$$
The integer number~$\omega$ may be called the {\it first specific
ordinal}, or simply the {\sc order}, and the other integer
number~$\omega'$ may be called the {\it second specific ordinal},
or simply the {\sc rank}, of the particular logarithmic function,
or {\it logarithm-couple}, which is determined by these two
integer numbers.  This existence of {\it two arbitrary and
independent integers in the general expression of a logarithm},
was discovered in the year 1826, by Mr.~{\sc Graves}, who
published a Memoir upon the subject in the Philosophical
Transactions for 1829, and has since made another communication
upon the same subject to the British Association for the
Advancement of Science, during the meeting of that Association at
Edinburgh, in 1834: and it was he who proposed these names of
{\it Orders and Ranks of Logarithms}.  But because
Mr.~{\sc Graves\/} employed, in his reasoning, the usual
principles respecting {\it Imaginary Quantities}, and was content
to prove the symbolical necessity without showing the
interpretation, or inner meaning, of his formul{\ae}, the present
{\it Theory of Couples\/} is published to make manifest that
hidden meaning: and to show, by this remarkable instance, that
expressions which seem according to common views to be merely
symbolical, and quite incapable of being interpreted, may pass
into the world of thoughts, and acquire reality and significance,
if Algebra be viewed as not a mere Art or Language, but as the
Science of Pure Time.  The author hopes to publish hereafter many
other applications of this view; especially to Equations and
Integrals, and to a Theory of Triplets and Sets of Moments,
Steps, and Numbers, which includes this Theory of Couples.

\bye

.
